A rocket is fired at a speed of 75.0 m/s from ground level, at an angle of 64.7¡ã above the horizontal. The rocket is fired toward an 11.0 m high wall, which is located 25.0 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?

Horizontal

distance=horzvelocity*time
time= distance/velocity= 25/(75cos64.7)
figure time, then put it here.
Vertical:
h=75*sin64.7t-4.9t^2
how much higher is h than 11 m.

thanks

where does the 4.9 come from?

h=49.62m

49.62m-11m for the wall
clear of the top = 38.62

4.9 its from multiplying 1/2(-9.8m/s^2)
1/2(ay)

To solve this problem, we can break it down into two components: the horizontal motion and the vertical motion.

First, let's find the time it takes for the rocket to reach the wall. We can use the horizontal component of the rocket's initial velocity.

Given:
Initial speed of the rocket (v0) = 75.0 m/s
Launch angle (θ) = 64.7 degrees
Horizontal distance to the wall (d) = 25.0 m

Horizontal component of velocity (v0x) = v0 * cos(θ)
v0x = 75.0 m/s * cos(64.7 degrees)
v0x = 75.0 m/s * 0.426
v0x = 31.95 m/s

Time taken to reach the wall (t) = d / v0x
t = 25.0 m / 31.95 m/s
t ≈ 0.782 seconds

Now, let's find the height reached by the rocket at this time.
We'll assume the acceleration due to gravity as 9.8 m/s^2.

Vertical component of velocity (v0y) at time t = v0 * sin(θ) - g * t
v0y = 75.0 m/s * sin(64.7 degrees) - (9.8 m/s^2) * 0.782 s
v0y = 63.80 m/s - 7.646 m/s
v0y ≈ 56.15 m/s

Now, we know the initial vertical velocity and the time. We can use the kinematic equation to calculate the height reached by the rocket.

Height reached (h) = v0y * t - (1/2) * g * t^2
h ≈ (56.15 m/s) * 0.782 s - (1/2) * (9.8 m/s^2) * (0.782 s)^2
h ≈ 43.92 m - 2.27 m
h ≈ 41.65 m

Finally, we need to subtract the height of the wall from this height to find how much the rocket clears the top.

Height cleared = h - 11.0 m
Height cleared ≈ 41.65 m - 11.0 m
Height cleared ≈ 30.65 m

Therefore, the rocket clears the top of the wall by approximately 30.65 meters.