a rocket blasts off and moves up with an acceleration of 12 m/s2 until it reaches 30 m then the engine shuts down and the rocket continues in free fall. What is the max hieght of the rocket and what is the total duration of the rockets flight?

To find the maximum height of the rocket, we need to use the kinematic equation for vertical motion:

h = (v^2 - u^2) / (2a)

where:
h = maximum height
v = final velocity
u = initial velocity
a = acceleration

In this case, the rocket has an initial velocity of 0 m/s (since it starts from rest on the ground) and an acceleration of 12 m/s^2 until it reaches 30 m. Therefore, we can substitute these values into the equation:

h = (v^2 - u^2) / (2a)
h = (0 - u^2) / (2a) (as the final velocity of the rocket at the maximum height is 0 m/s)
h = (0 - 0) / (2 * 12)
h = 0 / 24
h = 0 meters

Therefore, the maximum height of the rocket is 0 meters. This means that the rocket does not reach any higher point because the engine shuts down at 30 m.

To find the total duration of the rocket's flight, we need to calculate the time it takes for the rocket to reach 30 m using the equation:

v = u + at

where:
v = final velocity
u = initial velocity
a = acceleration
t = time

Given that the acceleration is 12 m/s^2, initial velocity is 0 m/s, and the final velocity is unknown, we can calculate the time it takes for the rocket to reach 30 m:

30 = 0 + 12t
12t = 30
t = 30 / 12
t = 2.5 s

Therefore, it takes 2.5 seconds for the rocket to reach a height of 30 meters.