A rocket is fired at a speed of 75.0 m/s from ground level, at an angle of 68.6° above the horizontal. The rocket is fired toward an 11.0 m high wall, which is located 26.0 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 of the rocket fired at speed v at angle a is given by

y(x) = -g*sec^2(a)/2v^2 x^2 + x*tan(a)
= -.0065x^2 + 2.55x

y(26) = 61.9m so it clears the 11-m wall by 50.9m

Makes sense, since if the rocket went in a straight line at an angle of 68.6°, at 26m out it would be 66.3m high.

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

Let's start by finding the time it takes for the rocket to reach the wall. We can use the horizontal motion formula:

Distance = Speed × Time

Given that the distance to the wall is 26.0 m, and the horizontal speed of the rocket is 75.0 m/s, we can rearrange the formula to solve for time:

Time = Distance / Speed

Time = 26.0 m / 75.0 m/s

Time ≈ 0.3467 s

Now, let's focus on the vertical motion of the rocket. We can use the vertical motion formula:

Vertical Distance = (Initial Vertical Speed × Time) + (0.5 × Acceleration × Time^2)

Since the rocket is launched at an angle above the horizontal, we need to calculate the initial vertical speed. We can use trigonometry to find it:

Initial Vertical Speed = Initial Speed × sin(θ)

Given that the initial speed is 75.0 m/s and the launch angle is 68.6°, we can calculate the vertical component of the initial speed:

Initial Vertical Speed = 75.0 m/s × sin(68.6°)

Initial Vertical Speed ≈ 67.3 m/s

Now, let's substitute the values into the vertical motion formula:

Vertical Distance = (67.3 m/s × 0.3467 s) + (0.5 × (-9.8 m/s^2) × (0.3467 s)^2)

Vertical Distance ≈ 23.4 m

Therefore, the rocket clears the top of the wall by approximately:

Vertical Distance - Height of the Wall

23.4 m - 11.0 m = 12.4 m

So, the rocket clears the top of the wall by about 12.4 meters.