A model rocket is launched upward with an initial speed of 50 m/s. It accelerates with a net constant upward acceleration of 2.0 m/s^2 until its engines stop at an altitude of 150 m. (a) What is the maximum height reached by the rocket? (b) how long after the life-off does the rocket reach its maximum height? (c) how long is the rocket in the air?

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To solve the given problem, we can use the equations of motion and kinematic equations. Let's break down each part of the question:

(a) What is the maximum height reached by the rocket?
To determine the maximum height reached by the rocket, we need to find the distance it travels while decelerating to a stop. We can use the kinematic equation:

v^2 = u^2 + 2as

Where:
v = final velocity (0 m/s since the rocket stops)
u = initial velocity (50 m/s)
a = acceleration (-2.0 m/s^2 since it is in the opposite direction)
s = distance traveled (unknown)

Rearranging the equation, we get:

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

Substituting the given values, we have:

s = (0^2 - 50^2) / (2 * -2)

Simplifying the equation, we get:

s = (0 - 2500) / -4
s = 2500 / 4
s = 625 m

Therefore, the maximum height reached by the rocket is 625 meters.

(b) How long after the lift-off does the rocket reach its maximum height?
To find the time it takes for the rocket to reach its maximum height, we need to use the equation:

v = u + at

Where the initial velocity (u) is 50 m/s, the acceleration (a) is -2.0 m/s^2, and the final velocity (v) is 0 m/s.

Substituting the values, we get:

0 = 50 + (-2)t

Rearranging the equation, we obtain:

-50 = -2t

t = -50 / -2
t = 25 seconds

Therefore, the rocket reaches its maximum height 25 seconds after lift-off.

(c) How long is the rocket in the air?
The total time the rocket is in the air is the time it takes to reach its maximum height plus the time it takes to fall back down to the ground. Since the rocket reaches its maximum height in 25 seconds and the fall back down is symmetrical, it will take an equal amount of time, making it 25 seconds.

Therefore, the rocket is in the air for a total of 50 seconds.