Amodel rocket is launched straight upward with an initial speed of 50.6 m/s. it accelerates with a constant upward acceleration of 2.74 m/s2 until its engines stop at an altitude of 160 m. what is the maximum height reached by the rocket? the acceleration of gravity is 9.81 m/s 2 . answer in units of m.

The maximum height reached by the rocket is 335 m.

Explanation:

The height of the rocket will be given by two equations:

y = y0 + v0 · t + 1/2 · a · t²    (while the engines are in function)

y = y0 + v0 · t + 1/2 · g · t² ( after the engines stop)

Where:

y = height at time t.

y0 = initial height.

v0 = initial velocity.

t = time.

a = acceleration due to the engines.

g = acceleration due to gravity (-9.81 m/s² considering the upward direction as positive)

In the same way, the velocity of the rocket will be given by two equations:

v = v0 + a · t

v = v0 + g · t

Where "v" is the velocity at time "t".

When the engines stop, the rocket is no more accelerated in the upward direction but it continues going up because it already has an upward velocity that will start decreasing until it becomes 0 m/s at the maximum height.

So, let´s find the velocity reached while the rocket was accelerated in the upward direction. For that, we need to know the time at which the engines stop, i.e., the rocket reached an altitude of 160 m.

y = y0 + v0 · t + 1/2 · a · t²

Since the origin of the frame of reference is located at the launching point, y0 = 0.

160 m = 50.6 m/s · t + 1/2 · 2.74 m/s² · t²

0 = 1.37 m/s² · t² + 50.6 m/s · t - 160 m

Solving the quadratic equation:

t = 2.93 s  

Now, using the equation of velocity:

v = v0 + a · t

v = 50.6 m/s + 2.74 m/s² · 2.93 s

v = 58.6 m/s

This velocity is the initial velocity of the rocket when it starts being accelerated downward due to gravity. When the rocket is at its maximum height, the velocity is 0. Then, we can calculate the time at which the rocket is at its max-height and with that time we can calculate the height:

v = v0 + g · t

0 = 58.6 m/s - 9.81 m/s² · t

-58.6 m/s / -9.81 m/s² = t

t = 5.97 s

Then, the maximum height will be:

y = y0 + v0 · t + 1/2 · g · t²

y = 160 m + 58.6 m/s · 5.97 s - 1/2 · 9.81 m/s² · (5.97 s)²

y = 335 m

The maximum height reached by the rocket is 335 m.

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