Interactive Dynamics
Projectile Motion


Leonardo da Vinci: An Artillery Park, c. 1487.

Introduction

Ballistics problems are a very important class of problems in engineering. A very precise and accurate understanding of projectile motion is needed in: Ballistics is, in essence, the study of projectile motion. Its development is very much related to the evolution of military technology and artillery in particular. One of the oldest problems that needed to be solved in the design of accurate cannons and rifles regards the interaction between a projectile and the medium in which it moves (for simplicity we will call air the fluid in question). Now you will confront this old problem using a new tool: video analysis! Unfortunately, the equations developed in class for projectile motion neglect enough important effects (e.g., air resistance, variations in the acceleration due to gravity, the size and shape of the object, the spin of the object, and more) to be of little use in real applications. Fortunately, we already understand enough to remedy much of this situation. In Part A of today's activity, we will analyze a situation in which our simple equations of motion do a superb job of accurately describing the motion of a projectile. This will also serve as an introduction to data collection in VideoPoint, data transfer to Excel, and then data analysis in Excel. Part B of this activity you will perform an analysis much like what you will do in Part A, but you will now include the effects of air resistance to solve a much more realistic problem.

Part A - Projectile Motion Without Air Resistance

In this activity you will use VideoPoint to digitize a QuickTime video to characterize the position of an object as a function of time. In particular, you will analyze the motion of a ball shot out of a small canon. Your initial objective is to collect the data for the horizontal and vertical positions as functions of time, i.e., x(t) and y(t), respectively and then to determine the velocity and accelerations as functions of time for the entire time in which the ball is in air.

You should begin by using VideoPoint to capture the position of the ball as a function of time from the movie entitled projectile.mov. Make sure that you have properly scaled your data from pixels to meters. Your time will be in seconds by default. After obtaining the data, copy it to Excel for analysis.

Your analysis should now include the following elements:

  1. A plot of the trajectory (y vs. x).

  2. A plot of the numerical approximation of the x-component of velocity and the y-component of velocity versus time. (See the Mini-Appendix on Numerical Derivatives.)

  3. A plot of the numerical approximation of the x-component of acceleration and the y-component of acceleration versus time.

  4. A comparison of your results from parts a, b, and c with the theoretical results. For example, you might address the following questions:
    1. Does your analysis of the data show that the acceleration of the object is always the acceleration due to gravity straight down?
    2. Is your velocity vector always tangent to the particle path?
    3. Is the path or trajectory of the ball a parabola?

    The more convincing your comparison, the better your report.

Mini-Appendix - Numerical Derivatives

You will need to compute numerical approximations of derivatives in this activity. Using Taylor series expansions of the derivatives with respect to time of the function f, it is not difficult to show that given the value of f at time , , and , then the first derivative of f at time t is:

and similarly, the second derivative of f with respect to time at time t is:


Part B - Projectile Motion With Air Resistance

Goals: To use the equations of motion to compute the air resistance coefficient.

Analysis

Here is a quick time movie (cartoon) showing the motion of two "cannon balls" shot up in the air. The spheres S1 and S2 shown in the movie are made of the same material and, have the same initial velocity. However, they differ in size. By studying their motion and by constructing a model for the air resistance force it is possible to calculate the air resistance coefficient.

In general, the resistance force due to solid-fluid interactions is assumed to depend on the velocity and geometry of the projectile in question. The air resistance is often assumed to be proportional to the projectile velocity when the projectile is moving slowly. This model of solid-fluid interaction is based on the physical assumption that the resistance offered by the fluid is, in essence, very similar to friction. However, as the projectile velocity increases, the character of the resistance changes and aerodynamic effects begin to significantly affect the projectile's motion. In this case, experimental evidence has shown that the fluid resistance is proportional to the square of the projectile's velocity and is also proportional to a coefficient that takes into account the projectile's geometry.

Your task is:

  1. to analyze the movie using VideoPoint and Microsoft Excel to determine establish which one of the two solid-fluid interaction models is best fits the motion of the two cannon balls;

  2. to propose and verify a model regarding the nature of the geometric coefficient mentioned above;

  3. finally, to compute the air resistance coefficient affecting the motion of the two cannon balls.
REMARK 1: the "number" used in the movie for the air resistance coefficient is not realistic. The choice of the air resistance coefficient was limited by the practical necessity of making a conveniently short quick time movie.

REMARK 2: from a geometrical viewpoint treat the cannon balls as disks one unit thick.

The following data concerning the cannon balls S1 and S2 will be useful in solving the problem:

Cannon Ball Properties Mass Radius
Ball S1 0.251 kg 0.1 m
Ball S2 1.005 kg 0.2 m


Activity Report

In your report you will provide a summary of the problem statement and its objectives. You should provide a description of your comparison (with all supporting calculations and figures) from Part A as the first part of your report. For the rest of the report, then you are required to described the two models of air resistance mentioned above and to describe what strategy you have used in selecting the appropriate model. Finally you are required to indicate the value you computed for the air resistance coefficient and to describe the strategy you used to determine it.



[EMch 12] [ESM] [Faculty] [PSU Engineering] [Penn State]


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Prepared by Gary L. Gray and Francesco Costanzo.
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© Copyright 1998 by Gary L. Gray and Francesco Costanzo. All rights reserved.