STATEMENT OF THE PROBLEM

1. A projectile fired from a catapult moves according to the parametric equations


x(t) = (v0 cos a)t +x0,

y(t) = -1/2 g t^2 + (v0 sin a)t + y0



where (x0, y0) is the starting point of the projectile, v0 is the initial velocity, a is the angle of elevation of the launch of the projectile from horizontal, and g is the acceleration due to gravity (either 32 ft/sec^2 or 9.81 m/sec^2). The functions x(t) and y(t) give the coordinates of the projectile at time t in seconds. This model does not take into consideration air resistance.

The distance that a catapult fires a projectile depends on two factors: (1) the projectile's initial velocity and (2) the angle of elevation of the launch of the projectile from horizontal. Suppose you can give the projectile an initial velocity of 300 ft/sec. Investigate the effect of varying the angle of elevation from the horizontal on the horizontal range of the projectile. What angle maximizes this range?

2. You are at the base of a long, steep hill which has an angle of elevation from the horizontal measuring 60 degrees. Before you lies a level plane. A target on the plane lies 3,000 ft from where you stand. You decide to catapult a projectile set at 45 degrees angle of elevation of the launch of the projectile from horizontal. Your projectile has an initial velocity of 300 feet per second.


Because of a stream which runs across the plane, you can only advance the catapult 30 ft. Where should you put it in order to score a direct hit on the target?

3. How far can we put the target and have it still within range of our catapult? Justify your answer.