kinetic energy and potential energy problem solving

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Potential And Kinetic Energy Example Problem – Work and Energy Examples

Potential energy is energy attributed to an object by virtue of its position. When the position is changed, the total energy remains unchanged but some potential energy gets converted into kinetic energy . The frictionless roller coaster is a classic potential and kinetic energy example problem.

The roller coaster problem shows how to use the conservation of energy to find the velocity or position or a cart on a frictionless track with different heights. The total energy of the cart is expressed as a sum of its gravitational potential energy and kinetic energy. This total energy remains constant across the length of the track.

Potential And Kinetic Energy Example Problem

A cart travels along a frictionless roller coaster track. At point A, the cart is 10 m above the ground and traveling at 2 m/s. A) What is the velocity at point B when the cart reaches the ground? B) What is the velocity of the cart at point C when the cart reaches a height of 3 m? C) What is the maximum height the cart can reach before the cart stops?

The total energy of the cart is expressed by the sum of its potential energy and its kinetic energy.

Potential energy of an object in a gravitational field is expressed by the formula

where PE is the potential energy m is the mass of the object g is the acceleration due to gravity = 9.8 m/s 2 h is the height above the measured surface.

Kinetic energy is the energy of the object in motion. It is expressed by the formula

KE = ½mv 2

where KE is the kinetic energy m is the mass of the object v is the velocity of the object.

The total energy of the system is conserved at any point of the system. The total energy is the sum of the potential energy and the kinetic energy.

Total E = KE + PE

To find the velocity or position, we need to find this total energy. At point A, we know both the velocity and the position of the cart.

Total E = KE + PE Total E = ½mv 2  + mgh Total E = ½m(2 m/s) 2  + m(9.8 m/s 2 )(10 m) Total E = ½m(4 m 2 /s 2 ) + m(98 m 2 /s 2 ) Total E = m(2 m 2 /s 2 ) + m(98 m 2 /s 2 ) Total E = m(100 m 2 /s 2 )

We can leave the mass value as it appears for now. As we complete each part, you will see what happens to this variable.

The cart is at ground level at point B, so h = 0 m.

Total E = ½mv 2  + mgh Total E = ½mv 2  + mg(0 m) Total E = ½mv 2

All of the energy at this point is kinetic energy. Since total energy is conserved, the total energy at point B is the same as the total energy at point A.

Total E at A = Total Energy at B m(100 m 2 /s 2 ) = ½mv 2

Divide both sides by m 100 m 2 /s 2 = ½v 2

Multiply both sides by 2 200 m 2 /s 2 = v 2

v = 14.1 m/s

The velocity at point B is 14.1 m/s.

At point C, we know only a value for h (h = 3 m).

Total E = ½mv 2 + mgh Total E = ½mv 2 + mg(3 m)

As before, the total energy is conserved. Total energy at A = total energy at C.

m(100 m 2 /s 2 ) = ½mv 2 + m(9.8 m/s 2 )(3 m) m(100 m 2 /s 2 ) = ½mv 2 + m(29.4 m 2 /s 2 )

Divide both sides by m

100 m 2 /s 2 = ½v 2 + 29.4 m 2 /s 2 ½v 2 = (100 – 29.4) m 2 /s 2 ½v 2 = 70.6 m 2 /s 2 v 2 = 141.2 m 2 /s 2 v = 11.9 m/s

The velocity at point C is 11.9 m/s.

The cart will reach its maximum height when the cart stops or v = 0 m/s.

Total E = ½mv 2 + mgh Total E = ½m(0 m/s) 2 + mgh Total E = mgh

Since total energy is conserved, the total energy at point A is the same as the total energy at point D.

m(100 m 2 /s 2 ) = mgh

100 m 2 /s 2  = gh

100 m 2 /s 2  = (9.8 m/s 2 ) h

The maximum height of the cart is 10.2 m.

A) The velocity of the cart at ground level is 14.1 m/s. B) The velocity of the cart at a height of 3 m is 11.9 m/s. C) The maximum height of the cart is 10.2 m.

This type of problem has one main key point: total energy is conserved at all points of the system. If you know the total energy at one point, you know the total energy at all points.

Related Posts

Potential and Kinetic Energy

Energy is the capacity to do work .

The unit of energy is J (Joule) which is also kg m 2 /s 2 (kilogram meter squared per second squared)

Energy can be in many forms! Here we look at Potential Energy (PE) and Kinetic Energy (KE).

Potential Energy and Kinetic Energy

• when raised up has potential energy (the energy of position or state)
• when falling down has kinetic energy (the energy of motion)

Potential energy (PE) is stored energy due to position or state

• a raised hammer has PE due to gravity.
• fuel and explosives have Chemical PE
• a coiled spring or a drawn bow also have PE due to their state

Kinetic energy (KE) is energy of motion

From PE to KE

For a good example of PE and KE have a play with a pendulum .

Gravitational Potential Energy

When the PE is due to an objects height then:

PE due to gravity = m g h

• m is the objects mass (kg)
• g is the "gravitational field strength" of 9.8 m/s 2 near the Earth's surface
• h is height (m)

Example: This 2 kg hammer is 0.4 m up. What is it's PE?

Kinetic energy.

The formula is:

KE = ½ m v 2

• m is the object's mass (kg)
• v is the object's speed (m/s)

Example: What is the KE of a 1500 kg car going at suburban speed of 14 m/s (about 50 km/h or 30 mph)?

KE = ½ × 1500 kg × (14 m/s) 2

KE = 147,000 kg m 2 /s 2

KE = 147 kJ

Let's double the speed!

Example: The same car is now going at highway speed of 28 m/s (about 100 km/h or 60 mph)?

KE = ½ × 1500 kg × (28 m/s) 2

KE = 588,000 kg m 2 /s 2

KE = 588 kJ

Wow! that is a big increase in energy! Highway speed is way more dangerous.

Double the speed and the KE increases by four times. Very important to know

A 1 kg meteorite strikes the Moon at 11 km/s. How much KE is that?

KE = ½ × 1 kg × (11,000 m/s) 2

KE = 60,500,000 J

KE = 60.5 MJ

That is 100 times the energy of a car going at highway speed.

When falling, an object's PE due to gravity converts into KE and also heat due to air resistance.

Let's drop something!

Example: We drop this 0.1 kg apple 1 m. What speed does it hit the ground with?

At 1 m above the ground it's Potential Energy is

PE = 0.1 kg × 9.8 m/s 2 × 1 m

PE = 0.98 kg m 2 /s 2

Ignoring air resistance (which is small for this little drop anyway) that PE gets converted into KE:

Swap sides and rearrange:

½ m v 2 = KE

v 2 = 2 × KE / m

v = √( 2 × KE / m )

Now put PE into KE and we get:

v = √( 2 × 0.98 kg m 2 /s 2 / 0.1 kg )

v = √( 19.6 m 2 /s 2 )

v = 4.427... m/s

Note: for velocity we can combine the formulas like this:

The mass does not matter! It is all about height and gravity. For our earlier example:

v = √( 2gh )

v = √( 2 × 9.8 m/s 2 × 1 m )

• Energy is the ability to do work

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Animated Physics Lessons

Mechanical Energy Problem Solutions

Mechanical energy problems and solutions.

See examples of mechanical energy problems involving kinetic energy, potential energy, and the conservation of energy. Check your work with ours.

1. How much gravitational potential energy do you have when you lift a 15 N object 10 meters off the ground?

2. How much gravitational potential energy is in a 20 kg mass when 0.6 meters above the ground?

3. How much gravitational potential energy does a 35 kg boulder have when 30 meters off the ground?

4. How many times greater is an objects potential energy when three times higher?

If you need help on ratio problems click the link below:

Rule of Ones: analyzing equations to determine how other variables change

5.  How much kinetic energy does a 0.15 kg ball thrown at 24 m/s have?

6.  How many times greater is the kinetic energy of a ball that is going five times faster?

7.  How much kinetic energy does a 1.2 kg ball have the moment it hits the ground 3.5 meters below when it starts from rest?

I cancelled out the initial kinetic energy because:

• KE i = ½ mv f 2
• KE i = (½)(3.5)(0 2 ) = 0 J

I cancelled out the final potential energy because:

• PE f = mgh f
• PE f = (3.5)(9.8)(0) = 0 J

8.  How fast is a 1.2 kg ball traveling the moment it hits the ground 3.5 meters below when it starts from rest?

(Note: In many of these problems I could cancel out mass but did not since it was provided)

Since I did not cancel out mass I could answer the following questions if asked:

• How much mechanical energy did you have at the beginning? (41.6 J)
• How much kinetic energy did you have at the beginning? (0 J)
• How much potential energy did you have at the beginning? (41.6 J)
• How much potential energy do you have at the end? (0 J)

If I cancelled out mass in my work it would not show the actual initial potential energy since PE i = mgh and not just gh.

9.  A 3.5 kg ball fell from a height of 12 meters.  How fast is it traveling when its still 5 meters off the ground?

10. An 85kg roller coaster cart is traveling 4 m/s at the top of a hill 50 meters off the ground.  How fast is it traveling at top of a second hill 20 meters off the ground?

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Kinetic energy problems

When solving kinetic energy problems, you may be asked to find 3 variables. These variables are the kinetic energy, the mass, or the speed.

Problem # 1:

Suppose a car has 3000 Joules of kinetic energy. What will be its kinetic energy if the speed is doubled? What if the speed is tripled?

We already proved in kinetic energy lesson that whenever the speed is doubled, the kinetic energy is quadrupled or four times as big.

4 × 3000 = 12000

Therefore, the kinetic energy is going to be 12000 joules.

Let v be the speed of a moving object. Let speed =  3v after the speed is tripled. 

9 × 3000 = 27000

Therefore, the kinetic energy is going to be 27000 joules.

Problem # 2:

Calculate the kinetic energy of a 10 kg object moving with a speed of 5 m/s. Calculate the kinetic energy again when the speed is doubled.

Tricky kinetic energy problems

Problem # 3: 

Suppose a rat and a rhino are running with the same kinetic energy. Which one do you think is going faster?

The only tricky and hard part is to use the kinetic energy formula to solve for v.

Multiply both sides by 2

Problem # 4: 

The kinetic energy of an object is 8 times bigger than the mass. Is it possible to get speed of the object?

Think carefully and try to solve this problem yourself.

Potential energy

Applied math

Calculators.

100 Tough Algebra Word Problems. If you can solve these problems with no help, you must be a genius!

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FREE K-12 standards-aligned STEM

curriculum for educators everywhere!

Find more at TeachEngineering.org .

• TeachEngineering
• Kinetic and Potential Energy of Motion

Lesson Kinetic and Potential Energy of Motion

Grade Level: 8 (7-9)

Time Required: 45 minutes

Lesson Dependency: None

Subject Areas: Physical Science, Physics

NGSS Performance Expectations:

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Curriculum in this Unit Units serve as guides to a particular content or subject area. Nested under units are lessons (in purple) and hands-on activities (in blue). Note that not all lessons and activities will exist under a unit, and instead may exist as "standalone" curriculum.

• Swinging Pendulum
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• Bouncing Balls: Collisions, Momentum & Math (for High School)
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TE Newsletter

Engineering connection, learning objectives, more curriculum like this, introduction/motivation, associated activities, lesson closure, vocabulary/definitions, user comments & tips.

Mechanical engineers are concerned about the mechanics of energy — how it is generated, stored and moved. Product design engineers apply the principles of potential and kinetic energy when they design consumer products. For example, a pencil sharpener employs mechanical energy and electrical energy. When designing a roller coaster, mechanical and civil engineers ensure that there is sufficient potential energy (which is converted to kinetic energy) to move the cars through the entire roller coaster ride.

After this lesson, students should be able to:

• Recognize that engineers need to understand the many different forms of energy in order to design useful products.
• Explain the concepts of kinetic and potential energy.
• Understand that energy can change from one form into another.
• Understand that energy can be described by equations.

Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN) , a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g. , by state; within source by type; e.g. , science or mathematics; within type by subtype, then by grade, etc .

Ngss: next generation science standards - science.

Common Core State Standards - Math

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International Technology and Engineering Educators Association - Technology

State standards, colorado - math, colorado - science.

Begin by showing the class three items: 1) an item of food (such as a bagel, banana or can of soda water), 2) a battery, and 3) you, standing on a stool or chair. Ask the class what these three things have in common. The answer is energy. The food contains chemical energy that is used by the body as fuel. The battery contains electrical energy (in the form of electrical, potential or stored energy), which can be used by a flashlight or a portable CD player. A person standing on a stool has potential energy (sometimes called gravitational potential energy) that could be used to crush a can, smash the banana, or really hurt the foot of someone standing under you. Do a dramatic demonstration of jumping down on the banana or an empty soda can. (Be careful! Banana peels are slippery!) Explain the ideas of potential energy and kinetic energy as two different kinds of mechanical energy . Give definitions of each and present the equations, carefully explaining each variable, as discussed in the next section,

PE = mass x g x height

KE = 1/2 m x v 2

Lesson Background and Concepts for Teachers

Whenever something moves, you can see the change in energy of that system. Energy can make things move or cause a change in the position or state of an object. Energy can be defined as the capacity for doing work. Work is done when a force moves an object over a given distance. The capacity for work, or energy, can come in many different forms. Examples of such forms are mechanical, electrical, chemical or nuclear energy.

This lesson introduces mechanical energy , the form of energy that is easiest to observe on a daily basis. All moving objects have mechanical energy. There are two types of mechanical energy: potential energy and kinetic energy. Potential energy is the energy that an object has because of its position and is measured in Joules (J). Potential energy can also be thought of as stored energy. Kinetic energy is the energy an object has because of its motion and is also measured in Joules (J). Due to the principle of conservation of energy, energy can change its form (potential, kinetic, heat/thermal, electrical, light, sound, etc.) but it is never created or destroyed.

Within the context of mechanical energy, potential energy is a result of an object's position, mass and the acceleration of gravity. A book resting on the edge of a table has potential energy; if you were to nudge it off the edge, the book would fall. It is sometimes called gravitational potential energy ( PE ). It can be expressed mathematically as follows:

PE = mass x g x height or PE = weight x height

where PE is the potential energy, and g is the acceleration due to gravity. At sea level, g = 9.81 meters/sec 2 or 32.2 feet/sec 2 . In the metric system, we would commonly use mass in kilograms or grams with the first equation. With English units it is common to use weight in pounds with the second equation.

Kinetic energy ( KE ) is energy of motion. Any object that is moving has kinetic energy. An example is a baseball that has been thrown. The kinetic energy depends on both mass and velocity and can be expressed mathematically as follows:

Here KE stands for kinetic energy. Note that a change in the velocity will have a much greater effect on the amount of kinetic energy because that term is squared. The total amount of mechanical energy in a system is the sum of both potential and kinetic energy, also measured in Joules (J).

Total Mechanical Energy = Potential Energy + Kinetic Energy

Engineers must understand both potential and kinetic energy. A simple example would be the design of a roller coaster — a project that involves both mechanical and civil engineers. At the beginning of the roller coaster, the cars must have enough potential energy to power them for the rest of the ride. This can be done by raising the cars to a great height. Then, the increased potential energy of the cars is converted into enough kinetic energy to keep them in motion for the length of the track. This is why roller coaters usually start with a big hill. As the cars start down the first hill, potential energy is changed into kinetic energy and the cars pick up speed. Engineers design the roller coaster to have enough energy to complete the course and to overcome the energy-draining effect of friction.

Watch this activity on YouTube

Restate that both potential energy and kinetic energy are forms of mechanical energy. Potential energy is the energy of position and kinetic energy is the energy of motion. A ball that you hold in your hand has potential energy, while a ball that you throw has kinetic energy. These two forms of energy can be transformed back and forth. When you drop a ball, you demonstrate an example of potential energy changing into kinetic energy.

Explain that energy is an important engineering concept. Engineers need to understand the many different forms of energy so that they can design useful products. An electric pencil sharpener serves to illustrate the point. First, the designer needs to know the amount of kinetic energy the spinning blades need in order to successfully shave off the end of the pencil. Then, the designer must choose an appropriately-powered motor to supply the necessary energy. Finally, the designer must know the electrical energy requirements of the motor in order for the motor to properly do its assigned task.

conservation of energy: A principle stating that the total energy of an isolated system remains constant regardless of changes within the system. Energy can neither be created nor destroyed.

energy: Energy is the capacity to do work.

kinetic energy: The energy of motion.

mechanical energy: Energy that is composed of both potential energy and kinetic energy.

potential energy: The energy of position, or stored energy.

Pre-Lesson Assessment

Discussion Questions: Solicit, integrate and summarize student responses.

• What are examples of dangerous unsafe placement of objects? (Possible answers: Boulders on the edge of a cliff, dishes barely on shelves, etc.).

Post-Introduction Assessment

Question/Answer: Ask the students and discuss as a class:

• What has more potential energy: a boulder on the ground or a feather 10 feet in the air? (Answer: The feather because the boulder is on the ground and has zero potential energy. However, if the boulder was 1 mm off the ground, it would probably have more potential energy.)

Lesson Summary Assessment

Group Brainstorm: Give groups of students each a ball (example, tennis ball). Remind them that energy can be converted from potential to kinetic and vice versa. Write a question on the board and have them brainstorm the answer in their groups. Have the students record their answers in their journals or on a sheet of paper and hand it in. Discuss the student groups' answers with the class.

• How can you throw a ball and have its energy change from kinetic to potential and back to kinetic without touching the ball once it relases from your hand? (Answer: Throw it straight up in the air.)

Calculating: Have students practice problems solving for potential energy and kinetic energy:

• If a mass that weighs 8 kg is held at a height of 10 m, what is its potential energy? (Answer: PE = (8 kg)*(9.8 m/s 2 )*(10 m) = 784 kg*m 2 /s 2 = 784 J)
• Now consider an object with a kinetic energy of 800 J and a mass of 12 kg. What is its velocity? (Answer: v = sqrt(2*KE/m) = sqrt((2 * 800 J)/12 kg) = 11.55 m/s)

Lesson Extension Activities

There is another form of potential energy, not related to height, which is called spring potential or elastic potential energy . In this case, energy is stored when you compress or elongate a spring. Have the students search the Internet or library for the equation of spring potential energy and explain what the variables in the equation represent. The answer is

PE spring = ½ k∙x 2

where k is the spring constant measured in N/m (Newton/meters) and x is how far the spring is compressed or stretched measured in m (meters).

This activity shows students the engineering importance of understanding the laws of mechanical energy. More specifically, it demonstrates how potential energy can be converted to kinetic energy and back again. Given a pendulum height, students calculate and predict how fast the pendulum will swing ...

This activity demonstrates how potential energy (PE) can be converted to kinetic energy (KE) and back again. Given a pendulum height, students calculate and predict how fast the pendulum will swing by understanding conservation of energy and using the equations for PE and KE.

High school students learn how engineers mathematically design roller coaster paths using the approach that a curved path can be approximated by a sequence of many short inclines. They apply basic calculus and the work-energy theorem for non-conservative forces to quantify the friction along a curve...

Students explore the physics exploited by engineers in designing today's roller coasters, including potential and kinetic energy, friction and gravity. During the associated activity, students design, build and analyze model roller coasters they make using foam tubing and marbles (as the cars).

Argonne Transportation - Laser Glazing of Rails. September 29, 2003. Argonne National Laboratory, Transportation Technology R&D Center. October 15, 2003. http://www.anl.gov/index.html

Asimov, Isaac. The History of Physics. New York: Walker & Co., 1984.

Jones, Edwin R. and Richard L. Childers. Contemporary College Physics. Reading, MA: Addison-Wesley Publishing Co., 1993.

Kahan, Peter. Science Explorer: Motion, Forces, and Energy. Upper Saddle River, NJ: Prentice Hall, 2000.

Luehmann, April. Give Me Energy. June 12, 2003. Science and Mathematics Initiative for Learning Enhancement, Illinois Institute of Technology. October 15, 2003. http://www.iit.edu/~smile/ph9407.html

Nave, C.R. HyperPhysics. 2000. Department of Physics and Astronomy, Georgia State University. October 15, 2003. hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

The Atoms Family - The Mummy's Tomb – Raceways. Miami Museum of Science and Space Transit Planetarium. October 15, 2003. http://www.miamisci.org/af/sln/mummy/raceways.html

Other Related Information

Browse the NGSS Engineering-aligned Physics Curriculum hub for additional Physics and Physical Science curriculum featuring Engineering.

Contributors

Supporting program, acknowledgements.

The contents of this digital library curriculum were developed under a grant from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education and National Science Foundation GK-12 grant no. 0338326. However, these contents do not necessarily represent the policies of the Department of Education or National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: December 14, 2022

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Kinetic Energy Problemset

SHOW ALL WORK!

1.  What is the kinetic energy of a jogger with a mass of 65.0 kg traveling at a speed of 2.5 m/s?

2.  What is the mass of a baseball that has a kinetic energy of 100 J and is traveling at 5 m/s?

Write down what you know:

3. What is the kinetic energy of a 0.5 kg soccer ball that is traveling at a speed of 3 m/s?

4  What is the kinetic energy of a 1 kg pie travelling at a speed of 4 m/s ?

5.  What is the kinetic energy of the pie if it is thrown at 10 m/s?

6. A student is hit with a 1 kg pumpkin pie. The kinetic energy of the pie 32 J. What was the speed of the pie?

GPE = mgh | g = 9.8 m/s 2

1.  Find the gravitational potential energy of a light that has a mass of 13.0 kg and is 4.8 m above the ground.

3.  A marble is on a table 2.4 m above the ground.  What is the mass of the marble if it has a GPE of 568 J. 

4.  A box with a mass of 12.5 kg sits on the floor.  How high would you need to lift it for it to have a GPE of 355J ?

5.  A cart at the top of a 300 m hill has a mass of  40 kg.  What is the cart’s gravitational potential energy?

6. Examine the graphic below.

What is the gravitational potential energy of the 6 kg cart as it sits the the top of the incline?  _______________

What is the KINETIC ENERGY of the cart if it is moving at a speed of 2 m/s at the bottom of the ramp? _____________

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Potential or Kinetic Energy?

8th -  11th  , potential & kinetic, 6th -  8th  , energy transformations, 4th -  7th  , gravitational potential energy, energy review, work, power, and energy quizizz, 11th -  12th  , kinetic and potential energy, potential and kinetic energy.

Potential and Kinetic Energy Problems

9th - 12th grade.

10 questions

Introducing new   Paper mode

No student devices needed.   Know more

The kinetic energy of a 7 kg cat moving at 4 m/s is

The gravitational potential energy of an object on the ground is

equal to the kinetic energy

The potential energy of a 550 kg missile flying at 800 meters is

4,312,000 Joules

440,000 Joules

176,000,000 Joules

A 80 kg runner going at 2.8 m/s has a kinetic energy of

2195.2 Joules

313.6 Joules

627.2 Joules

A ball that has a mass of 3kg and has a potential energy of 5,000 J is how high in the air.

A soccer ball has a gravitational potential energy of 80%. What is the percent of kinetic energy?

A student raises a 0.3 kg phone to a height of 0.8 meters. The potential energy of the phone is

.096 Joules

2.35 Joules

0.24 Joules

A plane with a mass 1200kg and a kinetic energy of 3,500,000 J is flying with a velocity of ?

5,833.3 m/s

What is the mass of a football that feel from a height of 28m with a potential energy of 4,150J?

A cannonball is launched with 500 Joules of kinetic energy and a velocity of 10m/s. What is its mass?

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7.2 Kinetic Energy and the Work-Energy Theorem

Learning objectives.

By the end of this section, you will be able to:

• Explain work as a transfer of energy and net work as the work done by the net force.
• Explain and apply the work-energy theorem.

Work Transfers Energy

What happens to the work done on a system? Energy is transferred into the system, but in what form? Does it remain in the system or move on? The answers depend on the situation. For example, if the lawn mower in Figure 7.2 (a) is pushed just hard enough to keep it going at a constant speed, then energy put into the mower by the person is removed continuously by friction, and eventually leaves the system in the form of heat transfer. In contrast, work done on the briefcase by the person carrying it up stairs in Figure 7.2 (d) is stored in the briefcase-Earth system and can be recovered at any time, as shown in Figure 7.2 (e). In fact, the building of the pyramids in ancient Egypt is an example of storing energy in a system by doing work on the system. Some of the energy imparted to the stone blocks in lifting them during construction of the pyramids remains in the stone-Earth system and has the potential to do work.

In this section we begin the study of various types of work and forms of energy. We will find that some types of work leave the energy of a system constant, for example, whereas others change the system in some way, such as making it move. We will also develop definitions of important forms of energy, such as the energy of motion.

Net Work and the Work-Energy Theorem

We know from the study of Newton’s laws in Dynamics: Force and Newton's Laws of Motion that net force causes acceleration. We will see in this section that work done by the net force gives a system energy of motion, and in the process we will also find an expression for the energy of motion.

Let us start by considering the total, or net, work done on a system. Net work is defined to be the sum of work on an object. The net work can be written in terms of the net force on an object. F net F net . In equation form, this is W net = F net d cos θ W net = F net d cos θ where θ θ is the angle between the force vector and the displacement vector.

Figure 7.3 (a) shows a graph of force versus displacement for the component of the force in the direction of the displacement—that is, an F cos θ F cos θ vs. d d graph. In this case, F cos θ F cos θ is constant. You can see that the area under the graph is F d cos θ F d cos θ , or the work done. Figure 7.3 (b) shows a more general process where the force varies. The area under the curve is divided into strips, each having an average force ( F cos θ ) i ( ave ) ( F cos θ ) i ( ave ) . The work done is ( F cos θ ) i ( ave ) d i ( F cos θ ) i ( ave ) d i for each strip, and the total work done is the sum of the W i W i . Thus the total work done is the total area under the curve, a useful property to which we shall refer later.

Net work will be simpler to examine if we consider a one-dimensional situation where a force is used to accelerate an object in a direction parallel to its initial velocity. Such a situation occurs for the package on the roller belt conveyor system shown in Figure 7.4 .

The force of gravity and the normal force acting on the package are perpendicular to the displacement and do no work. Moreover, they are also equal in magnitude and opposite in direction so they cancel in calculating the net force. The net force arises solely from the horizontal applied force F app F app and the horizontal friction force f f . Thus, as expected, the net force is parallel to the displacement, so that θ = 0º θ = 0º and cos θ = 1 cos θ = 1 , and the net work is given by

The effect of the net force F net F net is to accelerate the package from v 0 v 0 to v v . The kinetic energy of the package increases, indicating that the net work done on the system is positive. (See Example 7.2 .) By using Newton’s second law, and doing some algebra, we can reach an interesting conclusion. Substituting F net = ma F net = ma from Newton’s second law gives

To get a relationship between net work and the speed given to a system by the net force acting on it, we take d = x − x 0 d = x − x 0 and use the equation studied in Motion Equations for Constant Acceleration in One Dimension for the change in speed over a distance d d if the acceleration has the constant value a a ; namely, v 2 = v 0 2 + 2 ad v 2 = v 0 2 + 2 ad (note that a a appears in the expression for the net work). Solving for acceleration gives a = v 2 − v 0 2 2 d a = v 2 − v 0 2 2 d . When a a is substituted into the preceding expression for W net W net , we obtain

The d d cancels, and we rearrange this to obtain

This expression is called the work-energy theorem , and it actually applies in general (even for forces that vary in direction and magnitude), although we have derived it for the special case of a constant force parallel to the displacement. The theorem implies that the net work on a system equals the change in the quantity 1 2 mv 2 1 2 mv 2 . This quantity is our first example of a form of energy.

The Work-Energy Theorem

The net work on a system equals the change in the quantity 1 2 mv 2 1 2 mv 2 .

The quantity 1 2 mv 2 1 2 mv 2 in the work-energy theorem is defined to be the translational kinetic energy (KE) of a mass m m moving at a speed v v . ( Translational kinetic energy is distinct from rotational kinetic energy, which is considered later.) In equation form, the translational kinetic energy,

is the energy associated with translational motion. Kinetic energy is a form of energy associated with the motion of a particle, single body, or system of objects moving together.

We are aware that it takes energy to get an object, like a car or the package in Figure 7.4 , up to speed, but it may be a bit surprising that kinetic energy is proportional to speed squared. This proportionality means, for example, that a car traveling at 100 km/h has four times the kinetic energy it has at 50 km/h, helping to explain why high-speed collisions are so devastating. We will now consider a series of examples to illustrate various aspects of work and energy.

Example 7.2

Calculating the kinetic energy of a package.

Suppose a 30.0-kg package on the roller belt conveyor system in Figure 7.4 is moving at 0.500 m/s. What is its kinetic energy?

Because the mass m m and speed v v are given, the kinetic energy can be calculated from its definition as given in the equation KE = 1 2 mv 2 KE = 1 2 mv 2 .

The kinetic energy is given by

Entering known values gives

which yields

Note that the unit of kinetic energy is the joule, the same as the unit of work, as mentioned when work was first defined. It is also interesting that, although this is a fairly massive package, its kinetic energy is not large at this relatively low speed. This fact is consistent with the observation that people can move packages like this without exhausting themselves.

Example 7.3

Determining the work to accelerate a package.

Suppose that you push on the 30.0-kg package in Figure 7.4 with a constant force of 120 N through a distance of 0.800 m, and that the opposing friction force averages 5.00 N.

(a) Calculate the net work done on the package. (b) Solve the same problem as in part (a), this time by finding the work done by each force that contributes to the net force.

Strategy and Concept for (a)

This is a motion in one dimension problem, because the downward force (from the weight of the package) and the normal force have equal magnitude and opposite direction, so that they cancel in calculating the net force, while the applied force, friction, and the displacement are all horizontal. (See Figure 7.4 .) As expected, the net work is the net force times distance.

Solution for (a)

The net force is the push force minus friction, or F net = 120 N – 5 . 00 N = 115 N F net = 120 N – 5 . 00 N = 115 N . Thus the net work is

Discussion for (a)

This value is the net work done on the package. The person actually does more work than this, because friction opposes the motion. Friction does negative work and removes some of the energy the person expends and converts it to thermal energy. The net work equals the sum of the work done by each individual force.

Strategy and Concept for (b)

The forces acting on the package are gravity, the normal force, the force of friction, and the applied force. The normal force and force of gravity are each perpendicular to the displacement, and therefore do no work.

Solution for (b)

The applied force does work.

The friction force and displacement are in opposite directions, so that θ = 180º θ = 180º , and the work done by friction is

So the amounts of work done by gravity, by the normal force, by the applied force, and by friction are, respectively,

The total work done as the sum of the work done by each force is then seen to be

Discussion for (b)

The calculated total work W total W total as the sum of the work by each force agrees, as expected, with the work W net W net done by the net force. The work done by a collection of forces acting on an object can be calculated by either approach.

Example 7.4

Determining speed from work and energy.

Find the speed of the package in Figure 7.4 at the end of the push, using work and energy concepts.

Here the work-energy theorem can be used, because we have just calculated the net work, W net W net , and the initial kinetic energy, 1 2 m v 0 2 1 2 m v 0 2 . These calculations allow us to find the final kinetic energy, 1 2 mv 2 1 2 mv 2 , and thus the final speed v v .

The work-energy theorem in equation form is

Solving for 1 2 mv 2 1 2 mv 2 gives

Solving for the final speed as requested and entering known values gives

Using work and energy, we not only arrive at an answer, we see that the final kinetic energy is the sum of the initial kinetic energy and the net work done on the package. This means that the work indeed adds to the energy of the package.

Example 7.5

Work and energy can reveal distance, too.

How far does the package in Figure 7.4 coast after the push, assuming friction remains constant? Use work and energy considerations.

We know that once the person stops pushing, friction will bring the package to rest. In terms of energy, friction does negative work until it has removed all of the package’s kinetic energy. The work done by friction is the force of friction times the distance traveled times the cosine of the angle between the friction force and displacement; hence, this gives us a way of finding the distance traveled after the person stops pushing.

The normal force and force of gravity cancel in calculating the net force. The horizontal friction force is then the net force, and it acts opposite to the displacement, so θ = 180º θ = 180º . To reduce the kinetic energy of the package to zero, the work W fr W fr by friction must be minus the kinetic energy that the package started with plus what the package accumulated due to the pushing. Thus W fr = − 95 . 75 J W fr = − 95 . 75 J . Furthermore, W fr = f d ′ cos θ = – f d ′ W fr = f d ′ cos θ = – f d ′ , where d ′ d ′ is the distance it takes to stop. Thus,

This is a reasonable distance for a package to coast on a relatively friction-free conveyor system. Note that the work done by friction is negative (the force is in the opposite direction of motion), so it removes the kinetic energy.

Some of the examples in this section can be solved without considering energy, but at the expense of missing out on gaining insights about what work and energy are doing in this situation. On the whole, solutions involving energy are generally shorter and easier than those using kinematics and dynamics alone.

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/college-physics-2e/pages/1-introduction-to-science-and-the-realm-of-physics-physical-quantities-and-units

• Authors: Paul Peter Urone, Roger Hinrichs
• Publisher/website: OpenStax
• Book title: College Physics 2e
• Publication date: Jul 13, 2022
• Location: Houston, Texas
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• Section URL: https://openstax.org/books/college-physics-2e/pages/7-2-kinetic-energy-and-the-work-energy-theorem

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• Potential and Kinetic Energy

What is Energy?

Energy is the capacity to do work in physics. It exists in potential, kinetic, thermal, electrical, chemical, nuclear, or other forms. You can connect with the best online learning platform  Vedantu, to get a complete understanding of the topics related to the chapter Energy and download FREE PDF Potential and Kinetic Energy - Different Types, Formula, Solved Numericals, etc.

Potential Energy

Potential Energy definition states that “It is the energy stored that depends on the relative location of the different parts of the system”. In systems with parts that exert forces on each other of a magnitude depending on the configuration or relative position of the parts, Potential Energy arises.

When compressed or extended, spring has more Potential Energy. A steel ball after dropping to Earth has less Potential Energy compared to when lifted above the ground. Potential Energy is capable of doing more work in an elevated role. Potential Energy is a system property and not of an individual body or particle; for example, the system consisting of Earth and the raised ball has more Potential Energy as the two are separated further.

The Potential Energy of an object depends only on its original and final configurations. It is independent of the direction traveled by the objects. If the initial position of the ball is ground level and the final position is 10 m above the ground, in this case of the steel ball and the Earth, the Potential Energy is the same, regardless of how or by what route the ball was lifted. The value of Potential Energy is arbitrary and is proportional to the reference point selection. In the above example, if the initial location was the bottom of a 10 m deep pit, the device would have twice as much Potential Energy.

Different Types of Potential Energy

Electrical Potential Energy is the energy stored between the plates of a charged capacitor.

Chemical energy, the ability of a substance to work or to produce heat from a change in structure, may be considered as Potential Energy arising from the reciprocal forces between its molecules and atoms.

Nuclear energy is a form of Potential Energy as well.

Gravitational Potential Energy

The Potential Energy that a massive object has another massive object due to gravity is Gravitational Potential Energy. When the objects fall towards each other, it is the Potential Energy associated with the gravitational field that is released.

By multiplying the weight of an object by its distance above the reference point, Gravitational Potential Energy near the Earth's surface can be measured.

Inbound structures, such as atoms, where electrons are retained by the electrical force of attraction to nuclei, the zero reference for Potential Energy is such that the distance from the nucleus is not measurable by the electrical force. Bound electrons have negative Potential Energy in this case, and those so far away have zero Potential Energy.

Potential Energy Formula

The Gravitational Potential Energy formula relies on the force that acts on the two objects. The formula for the Gravitational Potential Energy is,

P.E. = m*g*h 

Where m is the mass in kilograms,

g is the acceleration due to gravity(9.8 m/s 2 at the earth's surface) and 

h is the height in meters.

The SI unit of measurement of Potential Energy is kg. m 2 /s 2 or Joule(J).

Some examples of Potential Energy include:

Compressed or extended spring.

A ball raised to some height.

Stored water in the Dam.

A car parked on the hilltop.

An arrow about to be shot.

Kinetic Energy

Kinetic Energy is the form of energy in which the object or a particle is said to be in motion. If the work that transfers energy is done on an object by applying a net force, the object speeds up and thus gains Kinetic Energy.

Kinetic Energy is a property of a moving object or particle which depends not only on its movement but also on its mass. Translation, rotation around an axis, vibration, or some combination of motions can be the form of Kinetic Energy.

Kinetic Energy Formula

Kinetic Energy is directly proportional to the object's mass and its velocity square, which is  K.E. = 1/2*m*v²

v is the velocity in m/s.

The SI unit of measurement of Kinetic Energy is the same as Potential Energy which is kg. m 2 /s 2 or Joule(J).

The Relation Between Potential Energy and Kinetic Energy

Kinetic Energy is nothing but a form of converted Potential Energy. Potential Energy can be transformed into the energy of motion such as Kinetic Energy and in turn into other forms, such as electric energy. Thus, through turbines that transform electric generators, water behind a dam flows to lower levels, generating electric energy plus some unusable heat energy resulting from turbulence and friction .

Potential Energy and Kinetic Energy are a form of mechanical energy so that the total energy in gravitational systems can be calculated as a constant.

Some examples of Kinetic Energy include:

Moving vehicle.

Running and Walking.

Bullet fired from a gun.

An arrow shot from a bow.

Solved Numerical on Potential and Kinetic Energy

1. In a running race competition, a student who is weighing 40 Kg is running at 4m/s. Calculate the Kinetic Energy of the student.

Ans: It is given that the weight/mass of the student, m = 40 Kg.

The velocity of a student, v = 4m/s.

Kinetic Energy is given by the formula, K.E.= 1/2*m*v²

Substituting the values we get, K.E.= 1/2 * 40 * 4*4 = 320 kg. m 2 /s 2 .

Therefore the Kinetic Energy of the student is 320 kg. m 2 /s 2 .

2. A water tank of mass 50 Kg is stored at a height of 10m. Calculate the Potential Energy of the tank. Consider the value of acceleration due to gravity(g) = 10 m/s 2 .

Ans: Given the mass of the tank, m = 50 Kg.

Height = 10m and g = 10 m/s 2 .

Potential Energy formula is given as P.E. = m*g*h.

Substituting the values we get P.E. = 50*10*10= 5000 kg. m 2 /s 2 .

Therefore the Potential Energy of the tank is 5000 kg. m 2 /s 2 .

FAQs on Potential and Kinetic Energy

1. What is Energy?

Energy in physics is the quantitative property that must be transferred to an object to operate on the object, or to heat it. For all living beings energy is the fundamental form of living. There are various types of energies. The element of energy on earth is the Sun. Energy is defined as the power or strength to do any kind of physical activity. Therefore it can be said that the ability to do work is known as Energy.

The Law of conservation of energy states that “the energy can neither be created nor be destroyed but it can only be converted from one form to another”. The standard unit of energy is Joule. Vedantu provides you with complete details about this topic to help you to understand it in a better way.

2. What is Potential Energy?

In physics, the energy retained by an object is Potential Energy regardless of its location relative to other objects, stresses within itself, its electrical charge, or other variables.

Types of Potential Energy are :

Gravitational Potential Energy: the energy stored in an object because of its height or vertical position. An  Example of Potential Energy can be a book kept on the higher shelf that has a higher level of gravitational Potential Energy than the book that is kept on the lower shelf.

Elastic Potential Energy: Energy stored because of the force applied to deform an elastic object is known as elastic Potential Energy. The energy that is formed because of the force is stored until the force is removed and the object recovers to its original shape. The object deformations can be of any form like compressing, stretching, or twisting the object.

Chemical Potential Energy: The energy stored in the chemical bonds of the substance is known as chemical Potential Energy. This energy is absorbed and released due to changes in the number of particles in the given species.

Electric Potential Energy the energy needed to move a charge against an electric field is known as electric Potential Energy. Examples of Potential Energy are :

A nonworking radio tower

A turned off black light

3. What is Kinetic Energy?

In physics, an object's Kinetic Energy is the energy it possesses due to its motion. It is defined as the work required to accelerate from rest to the specified velocity of a body of a given mass. Having accumulated this energy through its acceleration, until its speed increases, the body retains this Kinetic Energy.

Types of Kinetic Energy are :

Radiant Energy: the energy that travels by waves or particles is known as radiant energy, it is created through electromagnetic waves. Most humans experience this type of energy in the form of heat. Examples of Radiant Energy are: 

Two forms of energy are received when we turn on an incandescent light bulb, light and heat is generated which is visible; these are the two forms of radiant energies.

Sunlight is also radiant energy.

Thermal Energy: The energy experienced in the form of heat and warmth is called thermal energy. It is similar to radiant energy, thermal energy describes the level of activity among the atoms and molecules in an object but in radiant energy, we refer to waves and particles. Examples of thermal energy are :

While heating a pizza in the oven, we raise the temperature of the pizza. The molecule that makes up pizza moves faster when the pizza is piping hot. 

When the engine emits warmth it is an example of thermal energy.

Sound Energy: Sound is the vibrations that reach our ears. The vibrations move in the form of the wave through the medium of air to reach our eardrums. Once they reach then they are converted to electrical signals and then it is sent to the brain where we interpret the sensation of sound.

Electrical Energy: When around a circuit there is a flow of negatively charged electrons which results in electricity, which is known as electrical energy.

Mechanical Energy: the energy produced by the mechanical movements of the objects is called mechanical energy.

4. What is Energy Conversion?

Energy that is transferred and transformed is called conversion of energy. We all know that energy can be transferred from one form to another is called the transfer of energy. We can notice various energy transformations which are taking place around us.

Energy can be transferred in four ways :

By the action of force that is mechanically

By electricity that is electrically

By the mode of light waves or sound waves that is by radiation

By conduction, convection, or radiation that is by heating.

Energy transformation is the process by which the energy is changed from one form to another. The total amount of energy does not change when the energy is transferred or transformed, which is known as energy conservation.

5. Where can I find the best study material for chapter Energy?

Energy is a very important chapter which helps you to score good marks in exams. There are many important topics covered under this chapter which students should prepare well in order to get ready for the examination. 

To get the best study material for this chapter students can rely on Vedantu, which is the #1 online learning portal which provides you 100% accurate study material free of cost. You can easily download the free PDF for the revision notes and important questions of chapter Energy which are prepared by the expert team of Vedantu. The team of Vedantu has top-notch teachers that have years of experience to provide the best study material to students to help them to score well in examinations.

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KE = 0.5 • m • v 2 where m = mass of object

v = speed of object

Kinetic energy is a scalar quantity ; it does not have a direction. Unlike velocity , acceleration , force , and momentum , the kinetic energy of an object is completely described by magnitude alone. Like work and potential energy, the standard metric unit of measurement for kinetic energy is the Joule. As might be implied by the above equation, 1 Joule is equivalent to 1 kg*(m/s)^2.

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Check Your Understanding

Use your understanding of kinetic energy to answer the following questions. Then click the button to view the answers.

1. Determine the kinetic energy of a 625-kg roller coaster car that is moving with a speed of 18.3 m/s.

KE = (0.5) * (625 kg) * (18.3 m/s) 2

KE = 1.05 x10 5 Joules

2. If the roller coaster car in the above problem were moving with twice the speed, then what would be its new kinetic energy?

KE = 0.5*m*v 2

KE = 0.5*625 kg*(36.6 m/s) 2

KE = 4.19 x 10 5 Joules

3. Missy Diwater, the former platform diver for the Ringling Brother's Circus, had a kinetic energy of 12 000 J just prior to hitting the bucket of water. If Missy's mass is 40 kg, then what is her speed?

12 000 J = (0.5) * (40 kg) * v 2

300 J = (0.5) * v 2

600 J = v 2

v = 24.5 m/s

4. A 900-kg compact car moving at 60 mi/hr has approximately 320 000 Joules of kinetic energy. Estimate its new kinetic energy if it is moving at 30 mi/hr. (HINT: use the kinetic energy equation as a " guide to thinking .")

KE = 80 000 J

The KE is directly related to the square of the speed. If the speed is reduced by a factor of 2 (as in from 60 mi/hr to 30 mi/hr) then the KE will be reduced by a factor of 4. Thus, the new KE is (320 000 J)/4 or 80 000 J.

• Internal vs. External Forces

Kinetic Energy

Now, the bears I live with average, the males, eight to twelve hundred pounds [360 to 540 kg]. They're the largest bears in the world…. They've been clocked at 41 [mph] and they've run a hundred meter dash in 5.85 seconds, which a human on steroids doesn't even approach. Timothy Treadwell, 2001

• Compute the speed of a grizzly bear using Mr. Treadwell's hundred meter statement.
• Compute the kinetic energy of a grizzly bear using the speed you calculated in part a. and the average mass stated by Mr. Treadwell.
• How fast would a 250 lb man have to run to have the same kinetic energy you calculated in part b? (Do not use a calculator to compute your answer.)
• How fast would a 4000 lb car have to drive to have the same kinetic energy you calculated in part b? (Do not use a calculator to compute your answer.)

The Space Shuttle Columbia disintegrated during reentry on the morning of 1 February 2003. The cause of the accident was determined months later. A review of video footage taken during the launch 16 days earlier showed a large piece of foam insulation falling off the external fuel tank shortly after liftoff then striking the leading edge of the orbiter's left wing. This compromised the thermal protection system at the point of impact and allowed the superheated gases generated on reentry to melt the aluminum frame there. The left wing snapped off first, the orbiter tumbled and broke apart, scattering pieces across eastern Texas. All seven crew onboard were killed

Eighty-two seconds into STS 107 [the mission number], a sizeable piece of debris struck the left wing of the Columbia. Visual evidence and other sensor data established that the debris came from the bipod ramp area and impacted the wing on the wing leading edge. At this time Columbia was traveling at a speed of about 2,300 feet/second (fps) through an altitude of about 65,900 feet. Based on a combination of image analysis and advanced computational methods, the Board determined that a foam projectile with a total weight of 1.67 lb and impact velocity of 775 fps would best represent the debris strike…. Just prior to separating from the External Tank (ET), the foam was traveling with the orbiter at about 2,300 fps. The visual evidence shows that the debris impacted the wing approximately 0.161 seconds after separating from the ET. In that time, the debris slowed down from 2,300 fps to about 1,500 fps, so it hit the orbiter with a relative velocity of about 800 fps. In essence, the debris slowed down and the Orbiter did not, so that the Orbiter ran into the debris. Columbia Accident Investigation Board, 2003

Show that a piece of rigid foam insulation like the one that struck the Space Shuttle Columbia possesses a considerable amount of kinetic energy despite being "just a piece of foam".

• Determine the kinetic energy of the foam debris that struck Columbia in 2003.
• How fast would a 10 lb sledge hammer have to travel in order to have the same kinetic energy as the foam? State your answer in miles per hour or kilometers per hour as you prefer.
• How massive would a defensive tackle of American or Canadian football have to be if he ran as fast as a world class sprinter and had the same kinetic energy as the foam debris? State your answer in pounds or kilograms as you prefer.
• Write something different.
• Write something completely different.
• Is it possible for a motorcycle to have more kinetic energy than a truck?

Verify Robinson's first law of space combat (originally known as Robinson's first law of science fiction).

An object impacting at 3 km/s delivers kinetic energy equal to its mass in TNT. Ken Burnside, 2003

The term energy may be applied, with great propriety, to the product of the mass or weight of a body, into the square of the number expressing its velocity. Thus, if a weight of one ounce moves with the velocity of a foot in a second, we may call its energy 1; if a second body of two ounces have a velocity of three feet in a second, its energy will be twice the square of three, or 18. Thomas Young, 1807

Young would not receive full credit on an exam were he a student today. He provided units for the quantitites used in his calculations, but he neglected to include them in his solutions. Let young be the name of the unit that is missing from the passage above.

• How many youngs are in a joule (the unit of energy in the International System )?
• How many youngs are in a foot-pound (the unit of energy in the British Engineering System )?
• How many ergs (the unit of energy in the Gaussian System ) are in a young?

• tons of TNT (For comparison, the largest nuclear weapon ever tested had a yield of 50 million tons of TNT.)

Assuming that intensity is based on the kinetic energy of a "piece" of moving air, how many times more intense is…

• an EF2 than an EF1 tornado,
• an EF5 than an EF4 tornado,
• an EF5 than an EF1 tornado?

investigative

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Practice Problems on Potential Energy

In daily use, the potential word is used a lot for things or persons which show promise inside them. “Potential” shows the possibility for action. It gives an idea of stored energy that can be converted. This is the idea behind potential energy. This concept is an integral part of mechanics and allows us to theoretically measure the energy stored inside an object. Potential energy can come through any force. For example – a stretched or compressed spring has potential energy. An object sitting at some height possesses potential energy due to height. 

Potential Energy 

Potential energy is the energy possessed by an object due to its position or configuration. This energy is said to be stored inside the object. Usually, potential energy is released by an object by motion. For example, a stretched spring, when released, starts moving towards its natural position and starts acquiring speed. Due to this speed, it acquires kinetic energy. To further understand this, let’s consider a ball of mass “m”. 

This ball which was initially on the ground is taken to a height of “h”. External force in the form of gravity is acting on it. We know that work done by a force F on a displacement “s” on the object is given by, 

In this case, force is the force of gravity and displacement from the ground to the height “h”. 

F = mg, s = h 

Then, work done by gravity on the object is, 

Potential energy is defined as the negative of this work. Denoting the potential energy V(h). 

If the ball is dropped, in that case, the potential energy decreases, and the speed increases. This means the potential energy of the object is getting converted to kinetic energy. Let’s say the speed of the ball just before touching the ground is “v”. 

Potential Energy in a Spring

When the spring is kept normally, it is said to have 0 energy and is said to be in the equilibrium state. When it is stretched or compressed, and there is a certain displacement, say x, it will have certain potential energy saved in it which is given as,

P.E.= 1/2 (Kx 2 )

Where, K = Spring constant

x = displacement due to compression or expansion.

Let’s see some problems based on these concepts. 

Sample Problems

Question 1: A mass of 2Kg is taken from the ground to the height of 10m. Find the potential energy of the object.

The potential energy of a mass ‘m’ at the height ‘h’ is given by,  P = mgh  Given: m = 2kg and g = 10 m/s 2 and h = 10m.  Aim: Find the potential energy.  Plugging in the values in the formula.  P = mgh  ⇒ P = (2)(10)(10)  ⇒P = 200J Thus, the potential energy of the object is 200J. 

Question 2: A mass of 5Kg is taken from the ground to the height of 100m. Find the potential energy of the object.

The potential energy of a mass ‘m’ at the height ‘h’ is given by,  P = mgh  Given: m = 5kg and g = 10 m/s 2 and h = 100m.  Aim: Find the potential energy.  Plugging in the values in the formula.  P = mgh  ⇒ P = (5)(10)(100)  ⇒P = 5000J Thus, the potential energy of the object is 5000J. 

Question 3: A mass of 5Kg is taken from the ground for 5 m uphill on the wedge. The wedge makes an angle of 30° with the ground. Find the potential energy of the block. 

The potential energy of a mass ‘m’ at the height ‘h’ is given by,  P = mgh  This wedge is in the form of a right-angled triangle.  Figure  Let’s say, h is the vertical height at which the box reaches, let the slanted length be L L = 5 m  Given: m = 5kg and g = 10 m/s 2 and h = 2.5m.  Aim: Find the potential energy.  Plugging in the values in the formula.  P = mgh  ⇒ P = (5)(10)(2.5)  ⇒P = 125J Thus, the potential energy of the object is 125J.

Question 4: A mass of 10Kg is taken from the ground for 10m uphill on the wedge. The wedge makes an angle of 30° with the ground. Find the potential energy of the block. 

The potential energy of a mass ‘m’ at the height ‘h’ is given by,  P = mgh  This wedge is in the form of a right-angled triangle.  Figure  Let’s say, h is the vertical height at which the box reaches, let the slanted length be L L = 10m  Given: m = 5kg and g = 10 m/s 2 and h = 5 m.  Aim: Find the potential energy.  Plugging in the values in the formula.  P = mgh  ⇒ P = (5)(10)(5)  ⇒P = 250J Thus, the potential energy of the object is 250J.

Question 5: Find the kinetic energy of the ball just before hitting the ground. Assume that initially, the ball was at a height of 10m, and its mass was 2Kg. 

Answer: 

Initially, at the height of 10m, the ball possesses potential energy. When it is dropped, it starts going towards the ground and its height starts decreasing. With decreasing height, velocity increases, and it acquires kinetic energy.  Potential Energy at t = 0  Potential energy will be given by,  P = mgh m = 2Kg, h = 10m and g = 10 m/s 2 P = mgh  ⇒ P = (2)(10)(10)  ⇒P = 200J When the ball is about to hit the ground, it’s potential energy has become zero and all the energy is converted into kinetic energy.  Thus, K.E = 200J

Question 6: Find the velocity of the ball just before hitting the ground. Assume that initially, the ball was at a height of 100m, and its mass was 4Kg. 

Initially, at the height of 10m, the ball possesses potential energy. When it is dropped, it starts going towards the ground and its height starts decreasing. With decreasing height, velocity increases, and it acquires kinetic energy.  Potential Energy at t = 0  Potential energy will be given by,  P = mgh m = 4Kg, h = 100m and g = 10 m/s 2 P = mgh  ⇒ P = (4)(100)(10)  ⇒P = 4000J When the ball is about to hit the ground, its potential energy has become zero and all the energy is converted into kinetic energy.  Thus, K.E = 4000J The formula for K.E is,  K.E =  m = 4Kg and v = ?. Plugging the values in the formula K.E =  ⇒ 4000 =  ⇒2000 = v 2 ⇒ v = 10√20 m/s ⇒ v = 20√5 m/s

Question 7: The entire potential energy of a ball is transformed into its kinetic energy by coming on the ground from a certain height. The height at which the ball was initially placed was 10m. The mass of the ball is 1 kg. Find the gain in the kinetic energy.

Since, the all the potential energy present in the ball is transferred into its kinetic energy,  Potential energy of the ball = Final gain in the kinetic energy P= mgh m= 1kg, h= 10m, g= 9.8m/sec 2 P= 1× 10× 9.8  P= 98 Joule Therefore, the final gain in kinetic energy is 98 Joules. K.E= 98 Joule.

Question 8: Explain the existence of potential energy,

a. Due to its position

b. Due to the state in which the object is.

Potential energy can actually be present in two different cases, a. Due to its position Suppose an object is stable on the ground, now by applying some energy, it is taken at a certain height. The object present at a certain height will have energy saved in the form of potential energy. It is given as, P.E= mgh b. Due to the state in which object is present. When a spring is kept in normal state, it is said to have 0 energy, but when the same spring is either compressed or stretched, it obtains potential energy which is given as, P.E. = 1/2 (Kx 2 ) Where, x= displacement, K= Spring constant.

Question 9: A spring is stretched upto 9cm, the spring constant of the spring is 2 N/m. Find the value of Potential energy stored in the spring?

The potential energy stored in a spring is given as, P.E= 1/2 (Kx 2 ) K= 2 N/m, x= 9cm= 0.09m P.E= 1/2 (2× (0.09) 2 ) P.E= 8.1 × 10 -3 Joules.

Question 10: Two objects are kept at different heights, first object is at 5meters and second object is placed at 15 meters. Which object has more Potential energy if the first object is 5 times heavier than the second?

Object 1: Height= 5 m, Mass= 5 m P.E 1 = 5 m× 5× 9.8 P.E 1 = 245m Joules. Object 2: Height= 15 m, Mass= m P.E 2 = 15× m× 9.8 P.E 2 = 147m Joules. Therefore, even when the first object is kept at lesser height, due to its weight, the first object has more potential energy.

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Kinetic Energy: Explanation, Review, and Examples

• The Albert Team
• Last Updated On: February 16, 2023

We’ve learned a lot so far about how and why things move. What we haven’t learned is what allows one object to exert a force on another. Energy is an important concept in physics that is used to understand large and small processes. Here, we’ll begin with the energy of a moving object by exploring examples of kinetic energy and how to calculate it. We’ll also dive into the differences between kinetic and potential energy.

What We Review

What is Kinetic Energy?

“Kinetic” and “energy” are both words we should be familiar with at this point. “Kinetic” comes up when discussing the friction of a moving object. “Energy” is one you’ve probably heard a lot in your life. It probably has a meaning similar to “the ability to do something” and that’s what it means in physics, too. We refine it a bit to mean the ability to do work or exert a force on an object over a displacement. From this, we can guess that kinetic energy should have something to do with things moving and their ability to exert a force. While that isn’t a perfect definition, it is fairly close.

Kinetic energy does, indeed, have to do with an object in motion. It is the energy an object has because of its motion and we can think of it in terms of the force that can be exerted on an object or the force it would take to stop the object. The way we want to think about this force will change in different situations. In general, though, the kinetic energy of a moving object can be thought of as a measure of how far the object’s velocity is from zero. The greater the velocity, the greater the energy.

Examples of Kinetic Energy

One familiar example of kinetic energy and how it changes depending on velocity could be walking versus running. It’s much harder to stop quickly when running at full speed than it is when walking slowly. The same applies to driving a car – the faster it’s going, the sooner the brakes need to be hit. This applies to our everyday lives but is also often built into video games. Have you noticed how much harder it can be to take a sharp turn in a racing game when driving at full speed than it is when slowing down? You have kinetic energy to thank for that.

How to Calculate Kinetic Energy

So far, we’ve been talking a lot about the velocity of a moving object relative to kinetic energy. While that is an important factor, there is another factor – mass. The mass of an object is a measure of its inertia, of how hard it is to change its motion. To find the kinetic energy of an object, then, we must account for both its velocity and its mass. These two values don’t have the same impact on the kinetic energy of a moving object, though.

Kinetic Energy Formula

The formula for kinetic energy is written as:

Here, the E_{K} symbol is used to represent kinetic energy, m represents the mass, and lastly v represents the velocity. We’ll work with this equation in just a moment, but first, there are two very important conceptual points to be made here. First, is that we haven’t seen this combination of values before, which means we have a new unit to work with. Energy is measured in Joules, which is denoted J. Second, the velocity is squared while the mass is not.

How Mass and Velocity Affect Kinetic Energy

Because the velocity has a higher exponent than the mass does, the velocity will have a larger impact on the kinetic energy of a moving object. For example, if the mass were to double the kinetic energy would also double, but if the velocity doubled the kinetic energy would be quadrupled. Similarly, if the mass were halved then the kinetic energy would be halved, but if the velocity were halved the kinetic energy would be quartered.

Examples of Calculating Kinetic Energy

Now that we know the equation, we can start using it to calculate kinetic energy. We’ll follow our normal problem-solving method to first find kinetic energy and then to find mass from kinetic energy and velocity.

Example 1: Finding Kinetic Energy from Mass and Velocity

A 6\text{ kg} bowling ball is rolling toward the pins at 7\text{ m/s} . What is the kinetic energy of the bowling ball?

• m=6\text{ kg}
• v=7\text{ m/s}
• E_{K}=\text{?}
• E_{K}=\frac{1}{2}mv^{2}

Example 2: Finding Mass From Kinetic Energy and Velocity

A flamboyance of flamingo flies at an average speed of about 17\text{ m/s} . If one of the flamingos has a kinetic energy of 430\text{ J} , what is its mass?

• v=17\text{ m/s}
• E_{K}=430\text{ J}

Differences Between Kinetic and Potential Energy

Kinetic energy is not the only important energy type in mechanical physics. The other energy you’re likely to encounter is potential energy. Both kinetic energy and potential energy are important to how we view and understand the world and there is one key difference between them.

What is Potential Energy?

Potential energy is the energy stored in an object that gives it the potential to do something. This is the energy in an object that is not moving or exerting a force, but it could. We often see this as gravitational potential energy – the potential to fall – or as spring potential energy – the potential to launch something with a spring. A more in-depth explanation of potential energy can be found in a separate post , but the important thing to remember is that potential energy is the potential to complete an action.

How Kinetic and Potential Energy Relate to Each Other

Kinetic energy is energy that comes from motion. Potential energy, on the other hand, is the energy an object has the potential to use for motion. These energies can be transformed into one another. Let’s consider an example of potential energy transforming into kinetic energy. Before jumping out of a plane, a skydiver has no kinetic energy but does have a lot of potential energy. Once they jump out of the plane, their potential energy will decrease as they get closer to the ground and that energy will be shifting into kinetic energy as the skydiver’s velocity increases. The skydiver’s kinetic energy will increase at the exact same rate that its potential energy decreases. The same would be true for any object whose potential energy is transforming into kinetic energy or for one whose kinetic energy is transforming into potential energy.

Mechanical energy is constantly present in our lives and it’s an important part of how we understand the universe. Kinetic energy can be found in any moving object and there are examples of kinetic energy in many aspects of daily life. It is far from the only type of mechanical energy you’ll learn about on your physics journey, but knowing how to identify and calculate kinetic energy will help as you continue to expand your knowledge of how and why objects behave as they do.

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See how scores on each section impacts your overall ACT® score

Grammar Review Hub

Comprehensive review of grammar skills

AP® Posters

Download updated posters summarizing the main topics and structure for each AP® exam.