POSSIBLE SOLUTION(S)

1. (Restatement) 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?

Table Approach

We will use Mathematica to produce a table that shows how changing the barrel angle changes the range. Then we will study the table for conclusions about maximizing range. Basically this table is created by determining the time at which the projectile hits the ground (found in the command Solve[ y[t,theta]==0,t]) for given theta and then ascertains the x position (range) for that time.

This table indicates that range increases as theta increases, up to Pi/4, and then begins to decrease. The following graph, which plots range vs. angle, confirms that Pi/4, 45 degrees, does produce the maximum range. Again range function is produced by determining the x coordinate for the time t (Solve[ y[t,theta]==0,t]) at which the projectile strikes the ground.

We confirm the time of t = 13.2583 sec and range of 2812.5 ft for our trajectory with our angle of elevation a = Pi/4 .

Optimization Approach Using Calculus

2. (Restatement) 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 that plane 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?

We proceed upon the assumption that the angle of elevation with the horizontal remains Pi/4 radians or 45 degrees no matter if we are on level ground or up the hill we proceed to address this question.

Table approach

Optimization Approach Using Calculus

We now consider the more general case of moving the catapult up the hill AND changing the angle of elevation in order to reach the target. Perhaps we can determine a different angle which will still permit us to hit the target AND move the catapult up the hill less than when we fire at an angle of elevation of Pi/4 radians or 45 degrees.

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

Table approach

Optimization Approach Using Calculus

We now consider the more general case of moving the catapult up the hill AND changing the angle of elevation in order to reach the target. Perhaps we can determine a different angle which will still permit us to hit the target AND move the catapult up the hill less than when we fire at an angle of elevation of Pi/4 radians or 45 degrees.