A rocket is fired at a speed of 75.0 m/s from ground level, at an angle of 57.6¡ã above the horizontal. The rocket is fired toward an 11.0 m high wall, which is located 25.5 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?

1 m

To solve this problem, we can use the principles of projectile motion.

First, let's break down the initial velocity of the rocket into its horizontal and vertical components. The horizontal component will be given by:
Vx = V * cos(θ)
where V is the launch speed (75.0 m/s), and θ is the angle (57.6°) above the horizontal.

Vx = 75.0 m/s * cos(57.6°) = 40.40 m/s (rounded to two decimal places)

The vertical component will be given by:
Vy = V * sin(θ)

Vy = 75.0 m/s * sin(57.6°) = 64.71 m/s (rounded to two decimal places)

Next, let's determine the time it takes for the rocket to reach the top of its trajectory. Since the rocket starts and ends at the same vertical position, we know that it will take half the total time to reach its maximum height.

Using the formula for vertical displacement, we can find the maximum height (hmax) reached by the rocket:
hmax = (Vy^2) / (2 * g)
where g is the acceleration due to gravity (9.8 m/s^2)

hmax = (64.71 m/s)^2 / (2 * 9.8 m/s^2) = 215.32 m (rounded to two decimal places)

To find the time to reach the maximum height, we can use the formula:
Vy = gt
where t is the time in seconds.

64.71 m/s = 9.8 m/s^2 * t
t = 6.61 s (rounded to two decimal places)

Since the rocket reaches its maximum height in half the total time, the total time of flight (T) can be found by:
T = 2 * t
T = 2 * 6.61 s = 13.22 s (rounded to two decimal places)

Finally, to find the horizontal distance covered by the rocket, we can use:
x = Vx * T

x = 40.40 m/s * 13.22 s = 534.6 m (rounded to one decimal place)

Since the wall is located 25.5 m away, the rocket just clears the top of the wall with a distance of:
534.6 m - 25.5 m = 509.1 m (rounded to one decimal place)

Therefore, the rocket clears the top of the wall by approximately 509.1 meters.