A test rocket is fired straight up from rest with a net acceleration of 22.9 m/s2. After 8.8 seconds the motor turns off, but the rocket continues to coast upward with no appreciable air resistance. What maximum elevation does the rocket reach?

To solve this problem, we need to break it down into two parts: the ascent while the motor is on and the ascent while the rocket is coasting.

1. Ascent while the motor is on:

We can use the equation of motion: d = ut + 0.5at^2, where d is the distance, u is the initial velocity, a is the acceleration, and t is the time.

Here, u = 0 (since the rocket starts from rest), a = 22.9 m/s^2, and t = 8.8 s.

Substituting these values into the equation, we get:

d = 08.8 + 0.522.9*(8.8)^2 = 882.32 m

2. Ascent while the rocket is coasting:

When the motor turns off, the rocket continues to move upward due to its velocity at that moment. This velocity can be calculated using the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.

Here, u = 0, a = 22.9 m/s^2, and t = 8.8 s.

Substituting these values into the equation, we get:

v = 0 + 22.9*8.8 = 201.2 m/s

Now, the rocket will continue to ascend until its velocity becomes zero. We can find the distance it covers during this time using the equation v^2 = u^2 + 2ad, where v is the final velocity, u is the initial velocity, a is the acceleration (which is now -9.8 m/s^2 due to gravity), and d is the distance.

Here, v = 0, u = 201.2 m/s, and a = -9.8 m/s^2.

Rearranging the equation for d, we get:

d = (v^2 - u^2) / (2a) = (0 - (201.2)^2) / (2*-9.8) = 2065.47 m

So, the maximum elevation the rocket reaches is the sum of the distances covered in the two parts of its ascent:

882.32 m + 2065.47 m = 2947.79 m

Therefore, the rocket reaches a maximum elevation of approximately 2948 m.