Abstract

In physics class, you have probably rolled your eyes at some point after being assigned a "projectile motion" homework problem where you use equations to predict how a ball will move through the air. This experiment will show you just how fun that problem can be by using a real catapult to launch a ball and videotaping it as it flies along its path. Then, you will analyze the video and compare it to what the equations predicted. If you have ever wondered if those equations in your physics textbook are actually worth the paper they're printed on, this is your chance to find out.

Objective

Use projectile motion equations to predict the path of a ball launched through the air, then compare that path to the actual trajectory filmed during experiments.

Introduction

You have probably seen figures in your physics textbook that show a catapult launching a projectile and then equations that calculate the resulting trajectory. Well, it is a lot more fun if you actually get to use a catapult instead of just doing the calculations! In this science project, you will use a catapult to launch ping-pong balls and use a video camera to film their trajectory, or path, as they fly through the air. You can also use physics to predict the trajectory of the ball, and then compare this predicted trajectory to the one you measure from video recordings. If the results match, you can conclude that your predictions and the assumptions you used to make them were valid under the circumstances of the test. If they don't match exactly, that's OK. There are many real-world factors that can be difficult to account for in predictions. Part of the scientific process involves figuring out what those factors are so you can make better predictions next time.

Predicting the trajectory of a ball launched from the catapult requires an understanding of two fundamental physics concepts: projectile motion and conservation of energy.

Projectile Motion

Projectile motion refers to the method used for calculating the trajectory of a projectile (which can be pretty much any physical object — a rock, a ball, etc.) as it moves through the air.

Understanding projectile motion requires an understanding of position (the location of the ball, or projectile), velocity (how fast the ball is moving), and acceleration (how fast the velocity changes). A more advanced understanding of projectile motion involves calculus and factors like air resistance. For this project we will use simplified constant acceleration equations. These equations rely on the assumption that we can ignore air resistance — otherwise, the acceleration would change over time, and not be constant. (Note: Hover over the equations in this Introduction with your cursor to view enlarged formulas.) The four equations are:

Equation 1:

$$d_f = d_i + v_i t + \frac{1}{2}at^2$$

Equation 2:

$$v_f = v_i + at$$

Equation 3:

$$v_f^2 = v_i^2 + 2 a \left(d_f - d_i \right)$$

Equation 4:

$$d_f = d_i + \frac{1}{2}\left(v_f + v_i\right) t$$

In these equations, the subscripts i and f stand for initial and final, respectively. The variables d, v, and a stand for distance, usually expressed in meters (m); velocity, usually expressed in meters per second (m/s); and acceleration, usually expressed in meters per second squared (m/s²). The variable t stands for elapsed time, usually expressed in seconds (s).

Given certain information about an object at one time, these equations let you calculate what is happening at another time. For example, if you are running at an initial velocity of 3 m/s and accelerating at a rate of 1 m/s², you can use Equation 2 to calculate that your final velocity will be 5 m/s after a time (t = 2) seconds passed by:

$$v_f = 3\frac{m}{s} + 1\frac{m}{s^2}\cdot 2s = 5\frac{m}{s}$$

However, these equations are only for one dimension, and the catapult-launched ball in this experiment is going to be moving in two dimensions — x and y. Fortunately, we can take the same set of equations and apply them to the x and y directions independently — motion in the "y" direction does not affect motion on the "x" direction, and vice versa. Figure 1 shows a diagram of the setup for this experiment.

We want to calculate the position of the ball at any time, t. Thus we need equations for both x and y as functions of t. We will call these x(t) and y(t). Let us start with a new Equation 5 (using Equation 1), renaming the variables to be consistent with the x direction:

Equation 5:

$$x(t) = x_0 + v_{x,0}t+\frac{1}{2}a_x t^2$$

Two things here can immediately make this equation easier to work with: we can assume that the initial x position is just zero, so x₀=0. We also know that there is no acceleration in the x direction, because we are ignoring air resistance, and gravity certainly does not act sideways, so a_x=0. Now, Equation 5 reduces to:

Equation 6:

$$x(t) = v_{x,0}t$$

But we were only giving the initial velocity of the ball, v₀ , not the initial velocity in the x-direction, v_x,0 . However, we know the launch angle, θ. So, we can use trigonometry to calculate the initial x velocity:

Equation 7:

$$v_{x,0} = v_0 \cos(\theta)$$

Plug that into Equation 6, and we have our final equation for x as a function of time:

Equation 8:

$$x(t) = v_0 \cos(\theta) t$$

That is only half of the problem, though. We need to do the same thing for the y direction. Let us take Equation 1 again, this time substituting the appropriate y variables:

Equation 9:

$$y(t) = y_0 + v_{y,0}t + \frac{1}{2}a_y t^2$$

In this case, we cannot make any variables zero. We know from Figure 1 that the ball is launched from an initial height of h and that we have a downward (and therefore negative) acceleration of gravity, g. We can also calculate the initial velocity in the y direction, similar to what we did for the x direction in Equation 8, but this time using sine instead of cosine. Equation 9 thus becomes:

Equation 10:

$$y(t) = h + v_0 \sin(\theta) t - \frac{1}{2}g t^2$$

Now we can use Equations 8 and 10 to plot the motion of a projectile in the (x,y) plane.

Conservation of Energy

We are not finished yet, however — using Equations 8 and 10 requires that we know a few things — the height of the catapult above the ground, the angle at which the ball is launched, and its initial velocity. It is easy to measure the catapult's height with a tape measure, and the catapult in this kit makes it easy to measure the launch angle (more on that later). But what about the initial velocity of the ball? We cannot measure that directly, so we will use conservation of energy to estimate what it would be. In our case, we are dealing with kinetic energy (the energy an object with mass has according to its velocity) and potential energy (the energy that is stored in the rubber bands of the catapult). Note: We are ignoring a small amount of gravitational potential energy for now.

All types of energy are usually expressed in units of joules (or J). Conservation of energy states that, assuming losses like friction in a system are negligible, the sum of kinetic energy (or KE) and potential energy (or PE) always stays the same. This can be expressed in an equation:

Equation 11:

$$KE_i + PE_i = KE_f + PE_f$$

For this project, we will need to know the kinetic energy of a point mass (the ball or projectile can be approximated by a mass concentrated in a point), expressed as follows:

Equation 12:

$$KE = \frac{1}{2}mv^2$$

and the kinetic energy of a rotating object:

Equation 13:

$$KE = \frac{1}{2}I\omega^2$$

where m is the mass of an object (usually expressed in kilograms [kg]) and v its velocity; I the mass moment of inertia about the center of rotation (usually expressed in kilograms*meters squared [kg*m²]) and ω the angular velocity (expressed in radians per second [rad/s]).

We also need to determine the potential energy stored in a spring. Normally, for a linear spring, this would be expressed as:

Equation 14:

$$PE = \frac{1}{2}k\left(\Delta x\right)^2$$

where k is the spring constant (usually expressed in newtons per meter, [N/m]) and Δx is the displacement from the spring's rest position (do not confuse this with the distance x in Figure 1—they are different things). However, the rubber bands that come with the catapult kit do not behave this way — they act as nonlinear springs. The equation for their potential energy as a function of displacement Δx is:

Equation 15:

$$PE = \frac{2}{3}k \Delta x^{\frac{3}{2}}$$

This equation is specific to the rubber bands in this catapult kit and will not necessarily be true for every nonlinear spring. You can look in the Bibliography or any high-school physics textbook for more information on these topics. We have also included a Science Buddies reference page, Linear & Nonlinear Springs Tutorial if you would like to understand where Equations 14 and 15 come from.

Applying These Principles to the Catapult

The catapult kit is pictured in Figure 2, with all of its parts labeled. This catapult has two key features: an adjustable pin lets you set the launch angle (the angle in which the ball travels relative to horizontal at launch, shown in Figure 1) and tick marks let you measure the pull-back angle (how far back the arm is pulled before launch). Also, Figure 3 shows a zoomed-in view of how to measure these angles and Figure 4 labels them for clarity. The catapult kit also comes with two different balls (ping-pong and Wiffle®) and enables you to adjust the number of rubber bands. See the Bibliography for more information about the catapult kit.

Remember that our goal is to calculate the initial velocity of the ball, so we can then use the projectile motion equations to predict its path. To do this, we will compare the total energy of the catapult system in two different positions: when the arm is pulled back all the way, and immediately before the ball leaves the cup. These two positions are shown in Figure 5 with the following variables labeled:

• L is the length of the launch arm;
• r is the radius to the center of the ball (this is slightly smaller than L);
• x is the total length of the rubber band (it is folded in half for use on the catapult, so Figure 5 only shows ½ x); and
• v is the velocity of the ball.

Using Equations 11 through 14, we can write out the energy balance for these two positions. Three things can have energy in our system:

1. The ball can have kinetic energy (remember that we are ignoring gravity for now, since the elevation change is very small);
2. The arm of the catapult can have kinetic energy; and
3. The rubber bands can have potential energy.

The ball and the arm initially have zero velocity — and thus no kinetic energy. When you pull the arm back, potential energy is stored in the rubber bands. When you release the arm, the rubber bands contract, and their potential energy is converted to kinetic energy for the arm and the ball. The sum of these energies always has to be the same, which enables you to calculate the velocity of the ball. We will not go through the full derivation here — but advanced students may want to work this through and include it as part of their project. The energy balance, where N is the number of rubber bands, k is the spring constant of a single rubber band, x₀ is the unstretched length of a rubber band, m the mass of the ball, and I is the moment of inertia of the arm, is:

Equation 16:

$$\frac{2}{3}Nk \left( x_1 - x_0 \right) ^\frac{3}{2} = \frac{2}{3}Nk \left( x_2 - x_0 \right) ^\frac{3}{2} + \frac{1}{2}\left( m+ \frac{I}{r^2} \right) v_2^2$$

You will have to do some algebra to rearrange Equation 16 to solve for our unknown variable — the launch velocity of the ball, v₂. Also note that this equation makes use of the relationship between the linear velocity of the ball, v, and the angular velocity of the arm, ω:

Equation 17:

$$v = r \omega$$

Now that you know the launch velocity, you can go back to Equations 8 and 10, and you have everything you need to calculate the x and y positions of the ball at any time, t. This will give you your theoretical results or predicted trajectory — that is, what should happen according to equations you wrote down on paper. You will compare these to your experimental results — what actually happens when you launch the ball and measure its trajectory with recorded video.

A challenge for advanced students: The length of the rubber band, x, can just be measured with a ruler, but this could get annoying after a while. Can you use trigonometry to find a relationship between the length of the rubber band and the pull-back angle (which can be read directly from the disk on the catapult)?

Terms and Concepts

You should be familiar with the following terms and concepts before continuing with the project. We have provided a brief explanation of projectile motion and conservation of energy above, but you may need to consult resources in the Bibliography if you do not understand them.

• Trajectory
• Kinematics
• Projectile motion
  • Position
  • Velocity
  • Acceleration
• Constant acceleration equations
• Air resistance
• Conservation of Energy
  • Kinetic Energy
  • Potential Energy
• Spring constant
• Linear spring
• Nonlinear spring
• Launch angle
• Pull-back angle
• Theoretical results
• Experimental results

Questions

• How well do you think your theoretical results (also called predictions) will agree with your experimental results? Several things could affect this:
  • We ignored factors like gravitational potential energy and air resistance in the predictions. Do you think those will matter in the experiments, or will they be negligible?
  • There are several physical constants in the equations, like the spring constant of the rubber band, the mass of the ping pong ball, and the moment of inertia of the launch arm. Do you think small errors in the values for these constants could have a big impact on the theoretical results?
  • Your experimental results will depend on recorded videos of the ball's motion. Do you think you will be able to measure the ball's position in these videos accurately?
• If any of these things become problems, how could you address them to make your predictions or measurements more accurate?

Materials and Equipment

• Ping Pong Catapult Kit is available from our partner Home Science Tools. The kit includes:
  • Three rubber bands, 3 × 1/8 inches
  • Table tennis (ping pong) ball
  • Plastic ball with holes (like a Wiffle® ball)
  • Clamp for attaching the catapult to a surface
• Surface for mounting catapult (piece of wood on the floor, tabletop, etc.)
• Optional: paper towels, dish towel, or other padding for protecting the mounting surface from being scratched by the attachment clamp
• Open area for launching balls with bright lighting (either outside in direct sunlight, or inside with lots of lights)
• Video camera
  • A higher quality camera will give better experimental results.
  • A higher frame rate will also give better results (some cameras can film at 60 or 120 frames per second instead of just 30).
  • The ability to manually adjust the camera's exposure time will help your videos be less blurry — the ball moves very fast!
  • A tripod or other sturdy tool to mount and aim the camera is necessary — unless you have someone to help with a very steady hand!
• Ruler
• Media player program that enables you to step through videos frame-by-frame; Apple QuickTime® is a good option and can be downloaded for free.
• Microsoft Excel or other spreadsheet program
• Optional: a video tracking program like Tracker Video Analysis and Modeling, a free open-source video tracking program.
• Lab notebook

Experimental Procedure

Variations

Frequently Asked Questions (FAQ)

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