A rocket is fired at a speed of 50.0 m/s from ground level, at an angle of 46.0 ° above the horizontal. The rocket is fired toward an 18.7-m high wall, which is located 23.0 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?

To find out how much the rocket clears the top of the wall, we need to analyze the rocket's projectile motion.

First, we can break down the rocket's initial velocity into its horizontal and vertical components.

The horizontal component can be found using the equation:

Vx = V * cos(θ)

where Vx is the horizontal component, V is the launch speed (50.0 m/s in this case), and θ is the launch angle (46.0 ° in this case).

Let's calculate the horizontal component:

Vx = 50.0 m/s * cos(46.0 °)
Vx ≈ 50.0 m/s * 0.7193
Vx ≈ 35.97 m/s

Now, let's calculate the time it takes for the rocket to reach the wall. The time can be found using the equation:

t = d / Vx

where t is the time, d is the horizontal distance to the wall (23.0 m in this case), and Vx is the horizontal component of the velocity.

Let's calculate the time taken:

t = 23.0 m / 35.97 m/s
t ≈ 0.6399 s

Next, let's find out the rocket's vertical displacement during this time.

The vertical component of the initial velocity can be found using the equation:

Vy = V * sin(θ)

where Vy is the vertical component, V is the launch speed (50.0 m/s in this case), and θ is the launch angle (46.0 ° in this case).

Let's calculate the vertical component:

Vy = 50.0 m/s * sin(46.0 °)
Vy ≈ 50.0 m/s * 0.6947
Vy ≈ 34.74 m/s

Now, we can use the vertical motion equation to find the vertical displacement (clearance) during the time t:

h = Vy * t + (1/2) * g * t^2

where h is the vertical displacement (clearance), Vy is the vertical component of the velocity, t is the time, and g is the acceleration due to gravity (approximately 9.8 m/s²).

Let's calculate the vertical clearance:

h = 34.74 m/s * 0.6399 s + (1/2) * 9.8 m/s² * (0.6399 s)^2
h ≈ 22.19 m

Therefore, the rocket clears the top of the wall by approximately 22.19 meters.