A model rocket is launched straight upward with an initial speed of 30.0 m/s. It accelerates with a constant upward acceleration of 2.50 m/s2 until its engines stop at an altitude of 170 m.

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

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

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

Maximum height occurs where the velocity is temporarily zero.

While accelerating, the height is
y(t) = 30 t + 1.25 t^2
and the speed is
v(t) = 30 + 2.5 t

First figure out when y = 170m, and what the speed is at that time.

1.25 t^2 +30t -170 = 0
t^2 + 24t -136 = 0
That does not factor easily, so use the quadratic formula.
t = (1/2)(-24 + 33.47) = 4.73 s
v(t=4.73) = 30 + 11.83 = 41.83 m/s

Zero velocity will be reached after an additional time t', such that
g*t' = 41.83 m/s
t' = 4.27 s.
The total time after liftoff is then 9.00 seconds
The maximum altitude reached is
170 m + 41.83 t' -(g/2)t'^2
= 170 + 178.6 - 89.3 = 259.3 m

(b) t + t' = 9.00 s

(c) 9.00 s + (time to fall from 259.3 m)

To find the answers to these questions, we can use kinematic equations for motion with constant acceleration.

(a) To find the maximum height reached by the rocket, we need to find the time it takes for the rocket to reach its peak. We can use the equation:

v_f = v_i + at

where v_f is the final velocity, v_i is the initial velocity, a is the acceleration, and t is the time.

Since the rocket is moving upward, the final velocity at its maximum height will be 0 m/s. The initial velocity is 30.0 m/s and the acceleration is -2.50 m/s^2 (negative because it is acting in the opposite direction to the motion).

0 = 30.0 + (-2.50)t

Rearranging the equation, we have:

2.50t = 30.0

t = 30.0 / 2.50

t = 12.0 seconds

Now that we have the time it takes for the rocket to reach its maximum height, we can find the maximum height itself using the equation:

h = v_i * t + (1/2) * a * t^2

where h is the height.

h = 30.0 * 12.0 + (1/2) * (-2.50) * (12.0)^2

h = 360 - 180

h = 180 meters

Therefore, the maximum height reached by the rocket is 180 meters.

(b) The time it takes for the rocket to reach its maximum height is 12.0 seconds.

(c) The rocket is in the air for twice the time it takes to reach its maximum height since it takes the same amount of time to descend back to the ground.

Therefore, the rocket is in the air for 2 * 12.0 = 24 seconds.