CS190N/CS290N -- Assignment 2: Retracing the Path of a Bullet

John Brevik and Rich Wolski --- Winter, 2006

Your assignment is to investigate how much error their might be in the solution Charlie gives for finding of the first sniper we see in "Sniper Zero." After all, the sniper instructor makes it pretty clear that Charlie's estimate might need error bars (even though Charlie seems put out by the notion). Your investigation will comprise two parts.

Part A

You should implement four different ways of computing the height from which the bullet was fired given the range, impact angle, initial muzzle velocity, air density, bullet caliber, bullet ballistic coefficient, and bullet weight. The four methods, respectively, compute the initial height

if the shooter and target exist in a vacuum,
if there is an atmosphere and the drag force is proportional to the velocity,
if there is an atmosphere and the drag force is proportional to the square of the velocity using the Eulerian method of numerical integration, and
if there is an atmosphere and the drag force is proportional to the square of the velocity using Runge-Kutta-4 for numerical integration.

For all four methods, you should use the following set of given parameters:

range (horizontal distance): 900 feet
impact angle: 3 degrees
air density: 1.221 grams/liter
muzzle velocity: 2500 ft/s
bullet caliber: 270
ballistic coefficient: 0.392
bullet weight: 130 grains

For the linear-drag case, these parameters the constant drag factor equates to 0.778. In addition, you should also compute the initial height using each method for a different bullet and gun combination having the following characteristics:

muzzle velocity: 2900 ft/s
bullet caliber: 8.5 mm
ballistic coefficient: 0.565
bullet weight: 250 grain

which results in a constant drag factor of 0.44. You should also use a step size of 0.0001 and you should run your converging algorithms until the tolerance is under 0.001. You should also assume that the coefficient of drag is a linear function of bullet speed. For a standard "G1" bullet, the function you should use is C_d = 0.8365 - 0.0001094*s where s is the speed.

Part 2

In Charlie's "mind's eye" we see him attaching probabilities to the different possible locations from where the sniper might have fired. Assume that for each parameter, Charlie's value differs from the true value by an error term that is well-modeled by a normal distribution having a zero mean and a standard deviation that is 5% of the value Charlie chooses. Assume that all error probabilities are independent (e.g. Charlie's error in estimating the bullet weight is independent of the error he makes in estimating the muzzle velocity). Please provide an approximation to the 87% confidence interval for each of the 8 values you generated in Part A and justify your answer.

What to Turn In

You should turn in your code and the makefile needed to build it, the output you generated in answering Parts A and 2, a short written justification for your method of approximating the confidence interval, and a README file that explains any and all details necessary for the TA to be able to reproduce your results, and to test your code with a different set of parameters. As is the case for all assignments in this class, the TA must be able to build your program(s) using the default environment in the CSIL and to operate your program(s) in a way that blindingly straight forward.

Bonus Question -- What is Charlie's Processing Speed?

Charlie seems to make his estimates very quickly. If the time he takes to to calculate the 87% confidence interval for each possible location is one second, and he has to perform the same number of floating point computations as your program does, what is his processing rate (in megaflops per second) for the RK4 solution you provided in Part 2?

For the Graduate Student

If we had a handle on the final velocity of the bullet at impact, we could have solved the problem for both the range and height. Assume the the sniper used the first bullet configuration described in Part A and that the final speed is 2000 ft/s. Solve the system for both y₀ and x_f.