A test rocket is fired vertically upward from a well. A catapult gives it an initial speed of 81.0 m/s at ground level. Its engines then fire and it accelerates upward at 4.10 m/s2 until it reaches an altitude of 940 m. At that point its engines fail, and the rocket goes into free fall, with an acceleration of -9.80 m/s2. (You will need to consider the motion while the engine is operating separate from the free-fall motion.)

(a) How long is the rocket in motion above the ground?

(b) What is its maximum altitude?

(c) What is its velocity just before it collides with the Earth?

To solve this problem, we need to consider two separate motions of the rocket: the motion while the engine is operating and the motion during free fall.

(a) To determine the time the rocket is in motion above the ground, we can use the equations of motion.

During the engine operation, the rocket has an initial velocity (u) of 81.0 m/s, and it accelerates at 4.10 m/s^2 until it reaches an altitude of 940 m. We can use the following equation to find the time it takes to reach that altitude:

v^2 = u^2 + 2as

where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement. Plugging in the known values:

v^2 = (81.0 m/s)^2 + 2(4.10 m/s^2)(940 m)

v^2 = 6561 m^2/s^2 + 7672 m^2/s^2

v^2 = 14233 m^2/s^2

v = √(14233 m^2/s^2)

v ≈ 119.37 m/s

Now, we can use the equation of motion - v = u + at - to find the time it takes to reach that velocity:

v = u + at

119.37 m/s = 81.0 m/s + 4.10 m/s^2 * t

119.37 m/s - 81.0 m/s = 4.10 m/s^2 * t

38.37 m/s = 4.10 m/s^2 * t

t = 38.37 m/s / 4.10 m/s^2

t ≈ 9.37 s

Thus, the time the rocket is in motion above the ground during the engine operation is approximately 9.37 seconds.

(b) To find the maximum altitude, we need to determine the displacement covered during the engine operation. We can use the equation:

s = ut + (1/2)at^2

where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time. Plugging in the known values:

s = (81.0 m/s)(9.37 s) + (1/2)(4.10 m/s^2)(9.37 s)^2

s ≈ 381.1 m + 1788.6 m

s ≈ 2169.7 m

Thus, the maximum altitude the rocket reaches is approximately 2169.7 meters.

(c) When the rocket goes into free fall, its acceleration is -9.80 m/s^2. We can use the equation v = u + at to find the velocity just before it collides with the Earth. Since it is in free fall, the final velocity (v) will be 0 m/s and the initial velocity (u) will be the velocity right before the engines fail, which we found in part (a) to be approximately 119.37 m/s. The acceleration (a) is -9.80 m/s^2. Plugging these values into the equation:

0 m/s = 119.37 m/s + (-9.80 m/s^2)t

-119.37 m/s = -9.80 m/s^2 * t

t = -119.37 m/s / -9.80 m/s^2

t ≈ 12.18 s

Thus, the time it takes for the rocket to collide with the Earth after the engines fail is approximately 12.18 seconds.