objective of projectile motion experiment

5.3 Projectile Motion

Section learning objectives.

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

• Describe the properties of projectile motion
• Apply kinematic equations and vectors to solve problems involving projectile motion

Teacher Support

The learning objectives in this section will help your students master the following standards:

• (C) analyze and describe accelerated motion in two dimensions using equations.

In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Motion in Two Dimensions, as well as the following standards:

• (C) analyze and describe accelerated motion in two dimensions using equations, including projectile and circular examples.

Section Key Terms

Properties of Projectile Motion

Projectile motion is the motion of an object thrown (projected) into the air when, after the initial force that launches the object, air resistance is negligible and the only other force that object experiences is the force of gravity. The object is called a projectile , and its path is called its trajectory . Air resistance is a frictional force that slows its motion and can significantly alter the trajectory of the motion. Due to the difficulty in calculation, only situations in which the deviation from projectile motion is negligible and air resistance can be ignored are considered in introductory physics. That approximation is often quite accurate.

[BL] [OL] Review addition of vectors graphically and analytically.

[BL] [OL] [AL] Explain the term projectile motion. Ask students to guess what the motion of a projectile might depend on? Is the initial velocity important? Is the angle important? How will these things affect its height and the distance it covers? Introduce the concept of air resistance. Review kinematic equations.

The most important concept in projectile motion is that when air resistance is ignored, horizontal and vertical motions are independent , meaning that they don’t influence one another. Figure 5.27 compares a cannonball in free fall (in blue) to a cannonball launched horizontally in projectile motion (in red). You can see that the cannonball in free fall falls at the same rate as the cannonball in projectile motion. Keep in mind that if the cannon launched the ball with any vertical component to the velocity, the vertical displacements would not line up perfectly.

Since vertical and horizontal motions are independent, we can analyze them separately, along perpendicular axes. To do this, we separate projectile motion into the two components of its motion, one along the horizontal axis and the other along the vertical.

We’ll call the horizontal axis the x -axis and the vertical axis the y -axis. For notation, d is the total displacement, and x and y are its components along the horizontal and vertical axes. The magnitudes of these vectors are x and y , as illustrated in Figure 5.28 .

As usual, we use velocity, acceleration, and displacement to describe motion. We must also find the components of these variables along the x - and y -axes. The components of acceleration are then very simple a y = – g = –9.80 m/s 2 . Note that this definition defines the upwards direction as positive. Because gravity is vertical, a x = 0. Both accelerations are constant, so we can use the kinematic equations. For review, the kinematic equations from a previous chapter are summarized in Table 5.1 .

Where x is position, x 0 is initial position, v is velocity, v avg is average velocity, t is time and a is acceleration.

Solve Problems Involving Projectile Motion

The following steps are used to analyze projectile motion:

• Separate the motion into horizontal and vertical components along the x- and y-axes. These axes are perpendicular, so A x = A cos θ A x = A cos θ and A y = A sin θ A y = A sin θ are used. The magnitudes of the displacement s s along x- and y-axes are called x x and y . y . The magnitudes of the components of the velocity v v are v x = v ​ ​ ​ cos θ v x = v ​ ​ ​ cos θ and v y = v ​ ​ ​ sin θ v y = v ​ ​ ​ sin θ , where v v is the magnitude of the velocity and θ θ is its direction. Initial values are denoted with a subscript 0.
• Treat the motion as two independent one-dimensional motions, one horizontal and the other vertical. The kinematic equations for horizontal and vertical motion take the following forms Horizontal Motion ( a x = 0 ) x = x 0 + v x t v x = v 0 x = v x = velocity  is a constant. Horizontal Motion ( a x = 0 ) x = x 0 + v x t v x = v 0 x = v x = velocity  is a constant. Vertical motion (assuming positive is up a y = − g = − 9.80  m/s 2 a y = − g = − 9.80  m/s 2 ) y = y 0 + 1 2 ( v 0 y + v y ) t v y = v 0 y − g t y = y 0 + v 0 y t − 1 2 g t 2 v y 2 = v 0 y 2 − 2 g ( y − y 0 ) y = y 0 + 1 2 ( v 0 y + v y ) t v y = v 0 y − g t y = y 0 + v 0 y t − 1 2 g t 2 v y 2 = v 0 y 2 − 2 g ( y − y 0 )
• Solve for the unknowns in the two separate motions (one horizontal and one vertical). Note that the only common variable between the motions is time t t . The problem solving procedures here are the same as for one-dimensional kinematics.

Teacher Demonstration

Demonstrate the path of a projectile by doing a simple demonstration. Toss a dark beanbag in front of a white board so that students can get a good look at the projectile path. Vary the toss angles, so different paths can be displayed. This demonstration could be extended by using digital photography. Draw a reference grid on the whiteboard, then toss the bag at different angles while taking a video. Replay this in slow motion to observe and compare the altitudes and trajectories.

Tips For Success

For problems of projectile motion, it is important to set up a coordinate system. The first step is to choose an initial position for x x and y y . Usually, it is simplest to set the initial position of the object so that x 0 = 0 x 0 = 0 and y 0 = 0 y 0 = 0 .

Watch Physics

Projectile at an angle.

This video presents an example of finding the displacement (or range) of a projectile launched at an angle. It also reviews basic trigonometry for finding the sine, cosine and tangent of an angle.

• The time to reach the ground would remain the same since the vertical component is unchanged.
• The time to reach the ground would remain the same since the vertical component of the velocity also gets doubled.
• The time to reach the ground would be halved since the horizontal component of the velocity is doubled.
• The time to reach the ground would be doubled since the horizontal component of the velocity is doubled.

Worked Example

A fireworks projectile explodes high and away.

During a fireworks display like the one illustrated in Figure 5.30 , a shell is shot into the air with an initial speed of 70.0 m/s at an angle of 75° above the horizontal. The fuse is timed to ignite the shell just as it reaches its highest point above the ground. (a) Calculate the height at which the shell explodes. (b) How much time passed between the launch of the shell and the explosion? (c) What is the horizontal displacement of the shell when it explodes?

The motion can be broken into horizontal and vertical motions in which a x = 0 a x = 0 and   a y = g   a y = g . We can then define x 0 x 0 and y 0 y 0 to be zero and solve for the maximum height .

By height we mean the altitude or vertical position y y above the starting point. The highest point in any trajectory, the maximum height, is reached when   v y = 0   v y = 0 ; this is the moment when the vertical velocity switches from positive (upwards) to negative (downwards). Since we know the initial velocity, initial position, and the value of v y when the firework reaches its maximum height, we use the following equation to find y y

Because y 0 y 0 and v y v y are both zero, the equation simplifies to

Solving for y y gives

Now we must find v 0 y v 0 y , the component of the initial velocity in the y -direction. It is given by v 0 y = v 0 sin θ v 0 y = v 0 sin θ , where v 0 y v 0 y is the initial velocity of 70.0 m/s, and θ = 75 ∘ θ = 75 ∘ is the initial angle. Thus,

Since up is positive, the initial velocity and maximum height are positive, but the acceleration due to gravity is negative. The maximum height depends only on the vertical component of the initial velocity. The numbers in this example are reasonable for large fireworks displays, the shells of which do reach such heights before exploding.

There is more than one way to solve for the time to the highest point. In this case, the easiest method is to use y = y 0 + 1 2 ( v 0 y + v y ) t y = y 0 + 1 2 ( v 0 y + v y ) t . Because y 0 y 0 is zero, this equation reduces to

Note that the final vertical velocity, v y v y , at the highest point is zero. Therefore,

This time is also reasonable for large fireworks. When you are able to see the launch of fireworks, you will notice several seconds pass before the shell explodes. Another way of finding the time is by using y = y 0 + v 0 y t − 1 2 g t 2 y = y 0 + v 0 y t − 1 2 g t 2 , and solving the quadratic equation for t t .

Because air resistance is negligible, a x = 0 a x = 0 and the horizontal velocity is constant. The horizontal displacement is horizontal velocity multiplied by time as given by x = x 0 + v x t x = x 0 + v x t , where x 0 x 0 is equal to zero

where v x v x is the x -component of the velocity, which is given by v x = v 0 cos θ 0 . v x = v 0 cos θ 0 . Now,

The time t t for both motions is the same, and so x x is

The horizontal motion is a constant velocity in the absence of air resistance. The horizontal displacement found here could be useful in keeping the fireworks fragments from falling on spectators. Once the shell explodes, air resistance has a major effect, and many fragments will land directly below, while some of the fragments may now have a velocity in the –x direction due to the forces of the explosion.

[BL] [OL] [AL] Talk about the sample problem. Discuss the variables or unknowns in each part of the problem Ask students which kinematic equations may be best suited to solve the different parts of the problem.

The expression we found for y y while solving part (a) of the previous problem works for any projectile motion problem where air resistance is negligible. Call the maximum height y = h y = h ; then,

This equation defines the maximum height of a projectile . The maximum height depends only on the vertical component of the initial velocity.

Calculating Projectile Motion: Hot Rock Projectile

Suppose a large rock is ejected from a volcano, as illustrated in Figure 5.31 , with a speed of 25.0   m / s 25.0   m / s and at an angle 3 5 ° 3 5 ° above the horizontal. The rock strikes the side of the volcano at an altitude 20.0 m lower than its starting point. (a) Calculate the time it takes the rock to follow this path.

Breaking this two-dimensional motion into two independent one-dimensional motions will allow us to solve for the time. The time a projectile is in the air depends only on its vertical motion.

While the rock is in the air, it rises and then falls to a final position 20.0 m lower than its starting altitude. We can find the time for this by using

If we take the initial position y 0 y 0 to be zero, then the final position is y = − 20.0  m . y = − 20.0  m . Now the initial vertical velocity is the vertical component of the initial velocity, found from

Substituting known values yields

Rearranging terms gives a quadratic equation in t t

This expression is a quadratic equation of the form a t 2 + b t + c = 0 a t 2 + b t + c = 0 , where the constants are a = 4.90, b = –14.3, and c = –20.0. Its solutions are given by the quadratic formula

This equation yields two solutions t = 3.96 and t = –1.03. You may verify these solutions as an exercise. The time is t = 3.96 s or –1.03 s. The negative value of time implies an event before the start of motion, so we discard it. Therefore,

The time for projectile motion is completely determined by the vertical motion. So any projectile that has an initial vertical velocity of 14.3 m / s 14.3 m / s and lands 20.0 m below its starting altitude will spend 3.96 s in the air.

Practice Problems

The fact that vertical and horizontal motions are independent of each other lets us predict the range of a projectile. The range is the horizontal distance R traveled by a projectile on level ground, as illustrated in Figure 5.32 . Throughout history, people have been interested in finding the range of projectiles for practical purposes, such as aiming cannons.

How does the initial velocity of a projectile affect its range? Obviously, the greater the initial speed v 0 v 0 , the greater the range, as shown in the figure above. The initial angle θ 0 θ 0 also has a dramatic effect on the range. When air resistance is negligible, the range R R of a projectile on level ground is

where v 0 v 0 is the initial speed and θ 0 θ 0 is the initial angle relative to the horizontal. It is important to note that the range doesn’t apply to problems where the initial and final y position are different, or to cases where the object is launched perfectly horizontally.

Virtual Physics

Projectile motion.

In this simulation you will learn about projectile motion by blasting objects out of a cannon. You can choose between objects such as a tank shell, a golf ball or even a Buick. Experiment with changing the angle, initial speed, and mass, and adding in air resistance. Make a game out of this simulation by trying to hit the target.

Year 12 Physics Practical Investigation | Projectile Motion Experiment

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Sample Physics Practical Assessment Task: Projectile Motion Experiment

Projectile motion experiment is used by most schools for their first Physics practical assessment task. This is because most Projectile Motion practical investigation is relatively easy to design and conduct by students.

A typical Projectile Motion practical assessment task used by schools is outlined below.

In this sample practical assessment task, we are required to investigate the relationship between the range s_x and the launch velocity of a projectile released from an elevated position.

Let’s apply the scientific method to design and conduct a practical investigation for the assessment task outlined above.

Sample Physics Practical Report

The simplest type of projectile motion is a ball being projected horizontally from an elevated position.

In this situation, the range of a projectile is dependent on the time of flight and the horizontal velocity. Hence this experiment is based on the equation s_x=u_xt .

To express the time of flight t in terms of the acceleration due to gravity, we analyse the vertical motion of the projectile

s_y=u_yt+\frac{1}{2}at^2

-h=0+\frac{1}{2}(-g)t^2

t^2=\frac{2h}{g}

t=\sqrt{\frac{2h}{g}}

Hence the range of a projectile can be expressed in terms of the horizontal velocity and the other control variables such as y and g by substituting t=\sqrt{\frac{2h}{g}} expression into s_x=u_xt :

s_x = u_x \times (\sqrt{\frac{2h}{g}})

\therefore s_x = (\sqrt{\frac{2h}{g}}) u_x

2. Variables

Before designing your investigation, all the variables need to be identified.

• Independent variable: Horizontal launch velocity u_x
• Dependent variable: Range \Delta x
• Control variables: Height of the table y , acceleration due to gravity g , the shape of the projectile

Keeping the control variables constant allows the experiment to be more valid.

To learn more about how to improve the validity of your experiment, read the Matrix blog on ‘ Validity, Reliability and Accuracy of Experiments ‘

To determine the relationship between the range of a projectile \Delta x  and its horizontal launch velocity  u_x and use the results to calculate the acceleration due to gravity  g .

• A smooth metal ball is placed at the top of the ramp, and the vertical distance from the ball to the table is measured.
• The ball is rolled down and timed along the 1 \ m horizontal length using a stopwatch. The time is recorded.
• The distance from the foot of the table to its landing point on the carbon paper is observed, measured and recorded.
• Steps 1-4 are then repeated at different heights up the ramp.

The results are given in the table below. Using the times taken for the ball to travel 1 metre. Data collected from the experiment is highlighted in blue.

6. Quantitative Analysis of Results: Graphs and calculations

Calculate the horizontal velocity of the ball as it leaves the table and hence complete the table.

Plot the range of the ball \Delta x against the launch velocity u_x and draw in the line of best fit. 

• The range of the ball is plotted against the horizontal launch velocity.

• A line of best fit is drawn.

Determine the relationship between the launch velocity u_x and the range of the ball \Delta x  and hence discuss its significance

• The relationship between the launch velocity and the range of the ball is linear.  The range of the ball is directly proportional to the horizontal launch velocity: s_x = u_x \times t
• The linear relationship implies that the horizontal launch velocity affects the range but not the time taken to fall from a fixed height. Therefore horizontal and vertical motions are independent of each other.
• This also validates the results expected from the equations of projectile motion.

Use the gradient to find the acceleration due to gravity

7. Qualitative Analysis: Evaluation of method and errors

Let’s investigate the errors, reliability and accuracy of this experiment.

Access our library of Physics Practical Investigations.

Written by DJ Kim

DJ is the founder of Learnable and has a passionate interest in education and technology. He is also the author of Physics resources on Learnable.

Learnable Education and www.learnable.education, 2019. Unauthorised use and/or duplications of this material without express and written permission from this site's author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Learnable Education and www.learnable.education with appropriate and specific direction to the original content.

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• Newton's Laws
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• Newton's Laws of Motion
• Newton's First Law
• Newton's Third Law
• To know the definition of a projectile and to use concepts of force and inertia to explain the manner in which gravity affects a projectile.
• To be able to describe the horizontal and vertical components of the velocity of a projectile.
• To be able to describe the horizontal and vertical components of the displacement of a projectile.
• To be able to numerically describe the various features associated with a projectile’s trajectory (e.g., components
• To use kinematic equations to analyze and solve horizontally-launched projectile problems.
• To use kinematic equations to analyze and solve angle-launched projectile problems.

Readings from The Physics Classroom Tutorial

• What is a Projectile?  
• Characteristics of a Projectile's Trajectory  
• Horizontal and Vertical Components of Velocity  
• Horizontal and Vertical Displacement  
• Initial Velocity Components  
• Horizontally Launched Projectiles - Problem-Solving  
• Non-Horizontally Launched Projectiles - Problem-Solving

Interactive Simulations

Video and Animations

Labs and Investigations

• The Physics Classroom, The Laboratory, Basketball Analysis Students use video analysis to investigate the horizontal and vertical velocity and acceleration of a basketball.  
• The Physics Classroom, The Laboratory, Projectile Simulation Students use an online simulation to investigate the motion parameters of a projectile at a variety of locations along its trajectory.  
• The Physics Classroom, The Laboratory, Projectile Problem-Solving Students use an online application to master three types of horizontally-launched projectile problems. Students input answers and receive immediate feedback.  
• The Physics Classroom, The Laboratory, Launcher Speed Students fire a projectile launcher horizontally from a table top and make measurements in order to determine the launch speed of the projectile launcher.  
• The Physics Classroom, The Laboratory, Maximum Range Students use a projectile launcher to experimentally determine which angle projects a launched ball the furthest.  
• The Physics Classroom, The Laboratory, Hit the Target Students use a calibrated projectile launcher (from Lab 4: Launcher Speed above) and predict the initial height a target a known distance away must have in order for the launched projectile to strike the target. Link:   http://www.physicsclassroom.com/lab#vp

Demonstration Ideas

Minds On Physics Internet Modules:

• Vectors and Projectiles, Ass’t VP7 -  The Nature of a Projectile
• Vectors and Projectiles, Ass’t VP8 -  The Acceleration and Velocity of a Projectile
• Vectors and Projectiles, Ass’t VP9 -  Velocity Components for a Projectile
• Vectors and Projectiles, Ass’t VP10 -  Displacement and Time

Concept Building Exercises:

• The Curriculum Corner, Vectors and Projectiles, Projectile Motion Link: http://www.physicsclassroom.com/curriculum/vectors

Problem-Solving Exercises:

• The Calculator Pad, Vectors and Projectiles, Problems #21 - #34

Science Reasoning Activities:

• Science Reasoning Center, Vectors and Projectiles, Up and Down
• Science Reasoning Center, Vectors and Projectiles, Maximum Range of a Projectile
• Science Reasoning Center, Vectors and Projectiles, Juggling Link: http://www.physicsclassroom.com/reasoning/projectiles

Interactive Homework Problems

Real Life Connections:

Common Misconceptions:

• Horizontal Launches vs. Vertical Drops from the Same Height A common question that quickly uncovers a misconception is "If a ball is released from rest at the same time and from the same height that a second ball is launched horizontally, then which ball will strike the ground first." Quite surprising to students, the answer is that the balls strike the ground at the same time. Imparting an initial horizontal velocity to the second ball has no affect on its vertical motion. Perpendicular components of motion are independent of each other. The misconception uncovered by the leading question is that the two components of motion somehow depend upon one another. In students' minds, the changing of a horizontal parameter affects the vertical motion.  
• Horizontal Velocity Decreases with Time By definition, a projectile is an object upon which the only force is gravity. Gravity, being a vertical force, can only affect the vertical motion of a projectile. As such, the horizontal motion obtained at launch time does not change over the course of the motion. In the absence of horizontal forces, there is no horizontal acceleration for a projectile. The presence of air resistance would cause a launched object to decrease its horizontal velocity; but introducing air resistance into the discussion changes the topic from projectile motion to non-projectile motion.

Elsewhere on the Web:

• HS-PS2.1.i   Newton’s second law accurately predicts changes in the motion of macroscopic objects
• Algebraic thinking is used to examine scientific data and predict the effect of a change in one variable on another
• When investigating or describing a system, the boundaries and initial conditions of the system need to be defined.
• Analyze data using computational models in order to make valid and reliable scientific claims.
• Develop and use a model based on evidence to illustrate the relationships between systems or between components of a system.
• Use a model to provide mechanistic accounts of phenomena.
• Plan and conduct an investigation individually and collaboratively to produce data to serve as the basis for evidence … and consider limitations on the precision of the data
• Select appropriate tools to collect, record, analyze, and evaluate data.
• Collect data about a complex model or system to identify failure points or improve performance relative to criteria for success or other variables.
• Construct and revise an explanation based on valid and reliable evidence obtained from a variety of sources (including students’ own investigations, models, theories, simulations) and the assumption that theories and laws that describe the natural world operate today as they did in the past and will continue to do so in the future.
• Use mathematical representations of phenomena to support claims.
• Use mathematical representations of phenomena to describe explanations. Create or revise a computational model or simulation of a phenomenon, designed device, process, or system.
• Reason abstractly and quantitatively
• Model with mathematics
• Look for and express regularity in repeated reasoning
• N-VM.1      Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments and use appropriate symbols for vectors and their magnitudes.
• N-VM.2      Find the components of a vector.
• A-REI.4.b      Solve quadratic equations by inspection (e.g., for x squared = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation.
• A-REI.10      Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve.
• F-IF.4      For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
• F-IF.6      Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
• Use special triangles to determine geometrically the values of sine, cosine, tangent.
• RST.9-10.2     Determine the central ideas or conclusions of a text; trace the text’s explanation or depiction of a complex process, phenomenon, or concept; provide an accurate summary of the text.
• RST.9-10.3     Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.
• RST.11-12.4      Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 11-12 texts and topics.
• RST.11-12.7      Integrate and evaluate multiple sources of information presented in diverse formats and media in order to address a question or solve a problem.
• RST.11-12.8      Evaluate the hypotheses, data, analysis, and conclusions in a science or technical text, verifying the data when possible and corroborating or challenging conclusions with other sources of information.
• RST.11-12.9       Synthesize information from a range of sources (e.g., texts, experiments, simulations) into a coherent understanding of a process, phenomenon, or concept, resolving conflicting information when possible.)
• RST-9-10.10      By the end of Grade 10, read and comprehend science/technical texts in the grades 9-10 text complexity band independently and proficiently.

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Shop Experiment Projectile Motion Experiments​

Projectile motion.

Experiment #4 from Vernier Video Analysis: Motion and Sports

Introduction

Up to this point it is likely that you have examined the motion of an object in one dimension only—either falling vertically under the influence of Earth’s gravity or on a horizontal or inclined surface.

In this experiment, you will examine the behavior of a projectile—an object moving in space due to some initial launching force. Such an object can undergo motion in two dimensions simultaneously. Using the Vernier Video Analysis app, you will compare features of the position  vs . time and velocity  vs . time graphs with those of one-dimensional motion.

In this experiment, you will

• Use video analysis techniques to obtain position, velocity, and time data for a projectile.
• Analyze the position  vs . time and velocity  vs . time graphs for both the horizontal and vertical components of the projectile’s motion.
• Create and analyze your own video of an object undergoing projectile motion.

Sensors and Equipment

This experiment features the following sensors and equipment. Additional equipment may be required.

Correlations

Teaching to an educational standard? This experiment supports the standards below.

Ready to Experiment?

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Purchase the Lab Book

This experiment is #4 of Vernier Video Analysis: Motion and Sports . The experiment in the book includes student instructions as well as instructor information for set up, helpful hints, and sample graphs and data.