A rocket is launched from the origin with an acceleration of 20.0 m/s² at an angle of 30.0 degrees above the horizontal. The launch acceleration lasts for 2.00 seconds at which time the fuel is exhausted. The rocket then falls with an acceleration of 9.80 m/s² downward. What is the time it takes to reach maximum height? A) 3.22 s B) 3.87 s C) 4.04 s D) 4.75 s E) 5.12 s

Question

Quizplus · Accepted Answer

To find the time it takes for the rocket to reach maximum height, we need to use the kinematic equation:

y = yo + voy t + 1/2 a y t^2

where y is the height, yo is the initial height (which is zero since the rocket is launched from the ground), voy is the initial vertical velocity, ay is the vertical acceleration, and t is time.

We can break the rocket's motion into two parts: the upward motion while the rocket's engines are on, and the downward motion after the engines are off.

For the upward motion:
- Initial velocity in the upward direction: voy = 20 sin 30 = 10 m/s
- Vertical acceleration: ay = -9.80 m/s^2 (the negative sign is because the rocket is moving upward but decelerating)
- Time: t = 2.00 s (since the engines are on for 2.00 seconds)

Using the kinematic equation and plugging in these values, we can solve for the maximum height:

y = 0 + (10 m/s)(2.00 s) + 1/2 (-9.80 m/s^2)(2.00 s)^2
y = 10 m/s (2.00 s) - 19.6 m
y = -0.4 m

We see that the maximum height is negative, which makes sense since the rocket's engines can only propel it so high and it then falls back down. The negative sign indicates that the maximum height is 0.4 m below the initial height (i.e., the ground).

To find the time it takes to reach this maximum height, we need to use the same kinematic equation but solve for time instead of height:

y = yo + voy t + 1/2 a y t^2 
0 = 0 + 10 t + 1/2 (-9.80) t^2 
4.9 t^2 - 10 t = 0 
t(4.9 t - 10) = 0

The time t = 0 is not physically meaningful, so we need to choose the other solution:

t = 10 / 4.9 = 2.04 s

Therefore, the time it takes for the rocket to reach maximum height is 2.04 seconds, which is closest to choice C (4.04 s).

A Rocket Is Launched from the Origin with an Acceleration