**Virtual Simulations of Projectile Motion and Solutions to a System of Parametric Equations**

Nadina Duran-Hutchings, Ph.D.

C. Jay Hutchings, M.S.

**Abstract**

Galileo Galilei (1564-1642) [1] was the first researcher to analyze projectile motion in terms of the horizontal and vertical components separately, with the resulting parametric equations. He reached the conclusion that the path of any projectile is the result of two independent motions and the resulting path is that of a parabola, previously studied by the Greeks. Parametric equations are well suited for creating computer simulations of physical events. An Internet search for projectile motion will reveal many attempts by educators to build game like animations where students can practice launching projectiles at targets. None of the animations examined allowed the student to actually hit the targets accurately each time, and to earn points for the accuracy of the hits. This Macromedia Flash animation allows users to use a cannon to hit a bull's-eye every time, if they correctly solve the system of equations associated with the problem. In addition to allowing students to "do the math", special care was taken to provide a game like feel to the simulation. The intended goal is to provide an enjoyable and robust mathematics learning tool, suitable for class lectures and outside assignments in courses from College Algebra to Calculus. However, this game can also be used with a wider range of students for developing a qualitative understanding of the phenomenon under study, prior to the quantitative analysis.

The system that models the situation is:

**X = V cos (****q****) ***t* + X_{0}

**Y = Vsin (****q****) ***t* – 1/2g *t*^{2}+Y_{0}

The student needs to solve for *t* and then substitute the value into the second equation producing the general solution for **V**. The second equation calculates (X_{0}, Y_{0}) and is simplified somewhat differently.

The algebraic techniques needed to find the solution are not advanced. However some knowledge of trigonometry is needed to find Y_{0} and X_{0} with precision, although approximate values can be used. The player chooses a reasonable angle of elevation for the barrel and solves for V. If calculating errors are made the resulting answer generally appears unreasonable. Students can play the game and then print their results to turn in as an assignment.

The graph represents velocity solutions assuming that the initial cannonball and target positions are fixed. The initial position of the cannonball is dependent on the choice of angle. Therefore, once the angle has been selected the other parameters are fixed and velocity can be calculated. The velocity of a real cannonball is usually based upon how much gun powder is used and the mass of the ball. These relationships are non-linear.

The doted line model assumes the ball at the bottom of the barrel, or pivot point, and the solid line is the model for the ball at the tip of the barrel. The graph provides clear evidence that the 60 degree angle is an approximate minimal velocity solution for this particular choice of target location. The maximum horizontal range is achieved by an angle of 45 degrees as suggested in some physics textbooks [2]. The problem is more difficult when the vertical position of the target also changes. This raises a more complex question regarding the optimal choice of angle and initial velocity to allow the greatest probability of hitting the target under perturbed conditions.

The simulation uses the model that fixes the initial position at the tip of the cannon. Which is slightly more complicated because the location of the initial position of the ball is based upon the angle of elevation of the cannon. The alternative model simply establishes the initial position of the ball at the pivot point and, therefore, is constant. Both models perform equally well with angle values above 45 degrees and initial velocity greater than 100.

This Virtual Lab was designed to promote critical thinking skills and to provide the experience necessary for a student to move on to the more difficult task of modeling complicated physical phenomena quantitatively. The scoring method was introduced to tap into today's students ability with computer based games and to motivate them to find ways to achieve the highest score. The bullseye on the target is actually very small and only accurate shots are rewarded with the maximum ten points. Hitting the blue area scores two points and hitting the yellow area scores five points. Missing the target altogether results in a ten point deduction.

The Projectile Game provides the student with an opportunity to use different solutions based upon the imposed constraints of having to shoot over a brick wall. The level of difficulty increases as the number of cannonballs used increases since the brick wall increases in height. The simulation also has a calculator that will provide an initial velocity solution after the user enters their choice of angle of elevation. The user has to click on the copyright symbol at the bottom of the screen to bring up the calculator. Instructors may enable this feature for classroom demonstrations and have the print option disabled. For student assignments the print option should be enabled and the solution feature disabled.

**References**

- Wikipedia.org
- Giambattista, A., McCarthy-Richardson B., Richardson, R.; College Physics. McGraw Hill, 2004, page 124