Question

You have designed a rocket to be used to sample the local atmosphere for pollution. It is fired vertically with a constant upward acceleration of 17 m/s2. After 30 s, the engine shuts off and the rocket continues rising (in freefall) for a while. (Neglect any effects due to air resistance.) The rocket eventually stops rising and then falls back to the ground. You want to get a sample of air that is 23 km above the ground.

Answer 1

In order to get a sample of air that is 23 km above the ground, the rocket must rise by that height.

Draw a diagram (shown below) t represent the problem.

The distance h₁ is traveled from A to B with an upward acceleration of 17 m/s² over 30 s. The distance traveled is
h₁ = (1/2)(17 m/s²)(30 s)² = 7650 m = 7.65 km

At point B, the velocity attained is
V = (17 m/s²)30s) = 510 m/s

The distance h₂ is traveled from B to C with gravitational deceleration of -9.8 m/s². The distance traveled is given by
(510 m/s)² - 2(9.8 m/s²)h₂ = 0
h₂ = 13270.4 m = 13.27 km

The total height attained by the rocket is
h₁ + h₂ = 7.65 + 13.27 = 20.92 km

This height falls short of the desired height of 23 km.

Answer:
The rocket does not travel high enough to sample the air at a height of 23 km.