A new SpaceX rocket is being tested and launches from the ground. The first stage rocket engine provides constant upward acceleration during the burn phase. After the first stage engine burns out, the rocket has risen to 105.0 m and acquired an upward velocity of 50.0 m/s. The second stage fails to fire. The rocket continues to rise, reaches maximum height, and falls back to the ground. Neglect air resistance. The maximum height reached by the rocket is:

Question

A new SpaceX rocket is being tested and launches from the ground. The first stage rocket engine provides constant upward acceleration during the burn phase. After the first stage engine burns out, the rocket has risen to 105.0 m and acquired an upward velocity of 50.0 m/s. The second stage fails to fire. The rocket continues to rise, reaches maximum height, and falls back to the ground. Neglect air resistance. The maximum height reached by the rocket is:

A.256 m.

B.221 m.

C.244 m.

D.233 m.

209 m.

Answer 1

v^2 = 2gx

Solve for x and don't forget to add in the initial 105.

Answer 2

To find the maximum height reached by the rocket, we can use the equations of motion. In this scenario, the rocket experiences upward acceleration during its burn phase and then continues to rise after the first stage engine burns out.

Given data:
Initial position (s0) = 0 m
Final position (s) = 105.0 m
Initial velocity (v0) = 0 m/s
Final velocity (v) = 50.0 m/s
Acceleration (a) = constant upward acceleration during the burn phase

Using the equation of motion:

s = s0 + v0t + (1/2)at^2

Since the initial position (s0) and initial velocity (v0) are both 0, the equation simplifies to:

s = (1/2)at^2

We can rearrange this equation to solve for time (t):

t^2 = (2s)/a
t = sqrt((2s)/a)

Substituting the values:

t = sqrt((2 * 105 m) / a)

Now, we can use the equation of motion to find the maximum height reached by the rocket after the first stage engine burns out. This time, we assume the rocket has an initial position (s0) of 105.0 m and an initial velocity (v0) of 50.0 m/s (the velocity it acquired at the end of the first stage burn phase).

Considering the upward direction as positive, the equation becomes:

s = s0 + v0t + (1/2)at^2

Since we want to find the maximum height, the final velocity (v) at this point is 0 m/s. Hence:

0 = 105.0 m + 50.0 m/s * t + (1/2) * (-9.8 m/s^2) * t^2

Simplifying the equation:

0 = 105.0 m + 50.0 m/s * t - 4.9 m/s^2 * t^2

Now, we can substitute the value of t we found earlier:

0 = 105.0 m + 50.0 m/s * sqrt((2 * 105 m) / a) - 4.9 m/s^2 * (sqrt((2 * 105 m) / a))^2

Simplifying further:

0 = 105.0 m + 50.0 m/s * sqrt((2 * 105 m) / a) - 4.9 m/s^2 * (2 * 105 m) / a

Solving this equation will give us the value of a. Once we know the value of a, we can substitute it back into the equation to find the maximum height reached by the rocket.

Unfortunately, the question does not provide sufficient information to find the value of a, so it is not possible to determine the exact maximum height reached by the rocket.