Projectile Motion

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. You will investigate a launcher, and then make a mathematical model to predict where a projectile will land when the launcher is inclined to a specific launch angle. When you are satisfied with your model, you will use it to hit a target selected by your instructor.

There are three stages of this investigation:

  1. Investigate the launcher.
  2. Analyze 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.

Safety

The launching device is a spring gun. It is part of an apparatus called a ballistic pendulum. We will not be using the pendulum this week — only the launcher. The spring gun will fire a solid metal sphere at a moderate speed, so some precautions are in order.

  • Always make certain you have a clear path before launching a projectile.
  • Never load the spring gun until you are ready to fire.
  • Never walk in front of a loaded spring gun.

In other words, use common sense. Be certain no human is in the line of fire.

Investigate the Launcher

Your objective is to hit a target by adjusting the position and launch angle of the spring gun. As a first step toward this goal, you should investigate the following:

  • the range of a horizontal shot
  • the spread in impact points for “identical” initial conditions

To conclude this portion of the lab, use your knowledge of kinematics and the range of a horizontal shot to calculate the initial speed of the projectile as it leaves the launcher. Remember: When a projectile is fired horizontally, it moves with constant velocity in the horizontal direction and falls with constant acceleration in the vertical direction.

Investigate Range and Inclination

In this portion of the lab, you will gather data and model the range of the projectile as a function of launch angle. Before you begin taking measurements, adjust the location of your spring gun for maximum range. You should be able to hit points as close as \(1\,\mathsf{m}\) and as far as the barricade.

You will have to determine how to investigate a range of inclinations and how to interpret your data. (Part of experimental science and engineering is imagining what you could do if you had the ideal equipment. The more important part is accomplishing your objective with the materials available!)

Use your data to predict where the projectile will land for a given launch angle. Once you are confident in your model, practice hitting targets of your own. When you are satisfied, you may move on to the final stage.

Hit the Target

Your instructor will place a small target a fixed distance from the launcher. Your group will then adjust your launcher to hit the target.

View the target diagram

You must land one shot inside the target square full credit. If you hit the target using your first three shots, your group will earn 1 extra point toward your lab score. If you hit the bullseye, you will earn 2 points. However, for a given distance, you are not allowed to make any adjustments to the launcher after your first shot. After you fire the first shot, you have two options. You may fire two more shots without making any adjustments to the launcher; or, you may redo your calculations and try to hit a target placed at a different location.

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.

A Second Launcher: Repeat your study of initial velocity and model the relation between elevation and range for a second launcher. Compare this to the first. How consistent are the spring guns in the lab?

Slow Motion: Use slow-motion video and the Vernier Video Analysis software to determine the launch speed of your spring gun. Compare the result to your earlier analysis.

Long Shot: Compute the maximum range of the projectile based on your estimate of its initial speed. Collect data for the maximum range — in the hallway — and compare to your estimate.

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 \(V\).

Forces: Compute the spring constant of the spring gun. Explain your method and record any additional measurements you take.