A rocket is fired at a speed of 96.0 m/s from ground level, at an angle of 35.0 ° above the horizontal. The rocket is fired toward an 15.2-m high wall, which is located 32.0 m away. The rocket attains its launch speed in a negligibly short period of time, after which its engines shut down and the rocket coasts. By how much does the rocket clear the top of the wall?

the height h is

h(x) = x tanθ - g/(2(v cosθ)^2) x^2
= .7x - .0008 x^2
h(32) = 21.58

So, it clears the wall by 21.6-15.2 = 6.4m

940

To find out how much the rocket clears the top of the wall, we can break down the rocket's motion into horizontal and vertical components.

First, let's find the time it takes for the rocket to reach the wall using the horizontal component of its motion. The horizontal distance traveled by the rocket can be found using the formula: distance = speed * time. Since the speed of the rocket is 96.0 m/s and the distance to the wall is 32.0 m, we can rearrange the formula to solve for time: time = distance / speed.

time = 32.0 m / 96.0 m/s = 0.333 s

Next, let's find the maximum height reached by the rocket using the vertical component of its motion. The vertical distance traveled by the rocket can be found using the formula: distance = initial velocity * time + (1/2) * acceleration * time^2. Since the rocket starts at ground level, the initial vertical velocity is 0 m/s. The acceleration due to gravity is -9.8 m/s^2 (negative sign because it acts downwards). The time we found earlier is the time it takes for the rocket to reach the wall, so we can use this time in the formula.

distance = 0 * 0.333 s + (1/2) * (-9.8 m/s^2) * (0.333 s)^2
distance = -0.541 m (vertical distance below ground level)

Since the rocket starts from ground level and the vertical distance is negative, we can conclude that the rocket clears the top of the wall by:

Clearance = wall height - (vertical distance + height)

Clearance = 15.2 m - (-0.541 m + 0 m)
Clearance = 15.2 m + 0.541 m
Clearance ≈ 15.741 m

Therefore, the rocket clears the top of the wall by approximately 15.741 m.