Projectile Motion with Simulations

Objectives

In this lab, we will explore two-dimensional kinematics, the mathematical analysis of motion in two dimensions. Our focus will be the motion of a projectile. We will investigate a launcher, and then construct a mathematical model to predict where a projectile will land for a given launch angle, speed, and height. Finally, we will investigate how air resistance affects the range of a projectile.

There are three stages of this investigation:

  1. Investigate the range of a projectile.
  2. Analyze and predict projectile range as a function of launch angle.
  3. Hit a target selected and placed by your instructor.

You can — and should — apply your knowledge of kinematics in this lab.

Record your setup, procedure, data, and observations in your lab notes. Keep detailed notes. Data for this lab will include data tables and plots. In your memo, summarize your findings and describe how you tackled the design and measurement problems of this lab.

Horizontal Range

When a projectile is fired horizontally, it moves with constant velocity in the horizontal direction and falls with constant acceleration in the vertical direction. In this portion of the experiment, we will investigate the range of a projectile as a function of height and initial speed.

Start by opening the Projectile Motion simulation from PhET. Use the “Intro” option. This simulation gives you control over several factors that affect the range of a projectile. Play with the simulation to learn how it works.

Range vs. Height

Select an object and set the initial speed between 10 and \(20\,\mathsf{m/s}\). Adjust the launch angle to zero (horizontal). Turn off air resistance. Hold these parameters constant as you investigate the range of the projectile as a function of launch height.

Create a table and a plot of range vs. height. Record the range of the projectile for at least 5 different heights that span the range of 2 to \(15\,\mathsf{m}\).

Range vs. Speed

Select an object and set the initial height between 10 and \(20\,\mathsf{m}\). Adjust the launch angle to zero (horizontal). Turn off air resistance. Hold these parameters constant as you investigate the range of the projectile as a function of launch speed.

Create a table and a plot of range vs. speed. Record the range of the projectile for at least 5 different speeds that span the range of 0 to \(20\,\mathsf{m/s}\).

Analysis

Use kinematics to derive a formula for range \(D\) as a function of launch height \(H\) and launch speed \(V\) for a horizontal shot. Compare this function with your data.

Target Practice

It is possible to solve the kinematic equations for distance as a function of launch angle — as long as we ignore air resistance. However, in this portion of the lab, we will use interpolation to predict the range of a projectile given its launch angle. (Interpolation means using measured values of a function to predict values between the data points.)

Open the Projectile Motion simulation from PhET. Use the “Lab” option. This simulation gives you fine control over almost every factor that affects the range of a projectile. Play around with the new options to learn how they work.

Collect Your Data

Select an object and set the initial height to \(7\,\mathsf{m}\) and set the initial speed to \(17\,\mathsf{m/s}\). Turn off air resistance. Hold these parameters constant as you investigate the range of the projectile as a function of launch angle.

Create a table and a plot of range vs. launch angle. Record the range of the projectile for at least 5 different angles. Make sure the projectile range covers a wide range of distances from \(5\,\mathsf{m}\) to \(30\,\mathsf{m}\).

Use Your Data

Use your data to predict the angle required to hit the following target distances and then record the actual range of the projectile.

Do not use trial and error! You must explain how you used your data to predict each angle. You can use the plot, use linear interpolation, fit a mathematical function to your data, create a deep neural network, etc. The point of this exercise is to use your data to make the prediction. (If you are not sure where to begin, try the Wikipedia page for Linear Interpolation.)

Target Distance [m] Predicted Angle Actual Distance [m] Optimal Angle
8.4 m
18.2 m
29.0 m

After you have made your predictions and measurements — and only after — use trial and error to find the optimal launch angle. Discuss agreement or disagreement with your model in your memo.

Further Exploration

The following activities are not required. If you have the time and the inclination, you may carry out these investigations for extra credit. You are also free to use any of the materials available in the lab to carry out your own investigations. You need to keep a record of your work (setup, procedure, data, analysis) to get credit for your efforts.

Air Resistance: We have ignored air resistance in our analysis of projectile motion, but it can make a big difference. Use either the “Lab” or the “Drag” option of the simulation to explore the effects of air resistance.

Set the launch height, launch speed, and launch angle to give you a range of about \(20\,\mathsf{m}\) for the default projectile. Make a table and a plot of the range for one of the following parameters: mass, drag coefficient, elevation. Take at least 5 data points that cover the full range of allowed values. Discuss the relationship you observe. Is it linear or nonlinear? Can you think of a physical reason for the effect — e.g., why might the range depend on the elevation? You can repeat this analysis for all three factors, for up to 3 bonus points.

Theoretical Analysis: Derive the theoretical function that should fit your data from Part 2 of the experiment. I.e., use kinematics to derive a mathematical formula for distance as a function of angle \(D(\theta)\) for a given launch height \(H\) and speed \(V\).

Video Analysis: Use slow-motion video and the Vernier Video Analysis software to analyze the motion of a real projectile that you set in motion: a free throw, a baseball, any tossed object. Determine the launch speed of the projectile and its range.