A model rocket is launched straight upward with an initial speed of 30.0 m/s. It accelerates with a constant upward acceleration of 1.50 m/s2 until its engines stop at an altitude of 170 m.

(a) What is the maximum height reached by the rocket? (b) How long after lift-off does the rocket reach its maximum height(c) How long is the rocket in the air?

To find the answers to these questions, we can use the equations of motion for uniformly accelerated motion. The key equations we'll need are:

1. v = u + at (where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time)
2. s = ut + (1/2)at^2 (where s is the displacement)

Let's go through each question one by one:

(a) What is the maximum height reached by the rocket?

To find the maximum height, we need to determine the final velocity of the rocket when its engines stop. We know the initial velocity (u = 30.0 m/s) and the acceleration (a = 1.50 m/s^2), but we don't know the time it takes to reach the maximum height. We can use equation (1) to find the final velocity.

v = u + at
v = 30.0 m/s + (1.50 m/s^2)(t)
v = 30.0 m/s + 1.50 m/s^2(t)

At the maximum height, the final velocity will be zero. So, we can set v = 0 and solve for t:

0 = 30.0 m/s + 1.50 m/s^2(t)
-1.50 m/s^2(t) = -30.0 m/s
t = -30.0 m/s / -1.50 m/s^2
t = 20 s

Now that we have the time, we can use equation (2) to find the maximum height (s):

s = ut + (1/2)at^2
s = (30.0 m/s)(20 s) + (1/2)(1.50 m/s^2)(20 s)^2
s = 600 m + 300 m
s = 900 m

Therefore, the maximum height reached by the rocket is 900 meters.

(b) How long after lift-off does the rocket reach its maximum height?

From the previous calculation, we found that the time taken to reach the maximum height is 20 seconds.

(c) How long is the rocket in the air?

To find the total time the rocket is in the air, we need to consider both the ascent and descent phases. The time taken for ascent is the same 20 seconds calculated previously. Since the rocket is launched vertically upward and comes back down under gravity, it will take the same amount of time to come back down.

Therefore, the rocket is in the air for a total of 20 seconds for both ascent and descent phases.