Ballistics Project

Ballistics Project, (partial) Example Report
Name Here September 27, 2006

1

Overview

In this project we model the trajectories of a projectile launched at constant velocity and at angles ofinclination ranging from 5 to 85 degrees. Working initially under the assumption of no air resistance, we develop a simple Newtonian model for which trajectory paths can be computed explicitly.Comparing theoretical versus experimental plots of distance traveled versus angle of inclination, we find that this model fails to match our experimental data to a reasonable degree. As a first ordercorrection, we add linear air resistance, computing the coefficient of resistance from a separate experiment (rather than best fit of the data). Though significantly better, we find that our model involvinglinear air resistance still fails to adequately describe our experimental data. Last, we employ an argument based on dimension to determine that the force due to air resistance should depend on ....Incorporating this nonlinearity, we find that the mathematics becomes significantly more difficult, but that the resulting model fits our data quite well. We conclude that the primary forces on the projectile ofour experiment are gravity and ... air resistance, and that other effects are smaller perturbations of these.

2

Analysis in the Absence of Air Resistance
y ′′ (t) = −g, (1)

We begin ouranalysis in the absence of air resistance by using the height differential equation,

to find the initial velocity, v. Integrating (1) twice, we have y(t) = −gt2 /2 + vt + H, where g = 9.81 m/s2 and H =.39 m are the acceleration due to gravity and the initial height of our projectile respectively. Observing that y(2.13) = 0, we solve for v directly, obtaining the value v = 10.26 m/s. In order tostudy the trajectory of the projectile, we solve the system of differential equations y ′′ (t) = − g x′′ (t) = 0, with initial conditions y(0) = .18, y ′ (0) = v sin θ, x(0) = 0, x′ (0) = v cos θ. Again,...