To solve this problem, we can break it down into two parts: the motion of the rocket while the engine is operating and the free-fall motion after the engines fail. Let's solve it step by step:
(a) Motion while the engine is operating:
We can use the kinematic equation to find the time taken to reach the altitude of 920 m. The equation we need is:
h = ut + (1/2)gt^2
where h is the vertical displacement, u is the initial velocity, g is the acceleration due to gravity, and t is the time taken.
Given:
u = 80.8 m/s (initial speed)
g = 9.80 m/s^2 (acceleration due to gravity)
Setting the displacement h to 920 m and solving for t, we get:
920 = 80.8t + (1/2)(3.90)t^2
Simplifying and rearranging the equation:
0.195t^2 + 80.8t - 920 = 0
We can solve this quadratic equation to find the time taken to reach an altitude of 920 m. Using the quadratic formula, we get:
t = (-80.8 ± √(80.8^2 - 4(0.195)(-920))) / (2(0.195))
Solving this equation, we find two possible solutions for t:
t1 ≈ 7.51 s
t2 ≈ -48.26 s (this negative value is not valid)
Therefore, the rocket is in motion above the ground for approximately 7.51 seconds.
(b) Maximum altitude:
To find the maximum altitude, we need to calculate the displacement at that point using the equation:
h = ut + (1/2)gt^2
Using the given values:
u = 80.8 m/s (initial speed)
g = 9.80 m/s^2 (acceleration due to gravity)
t = 7.51 s (time when the rocket reaches its maximum altitude)
Substituting these values into the equation, we have:
h = (80.8)(7.51) + (1/2)(-9.80)(7.51)^2
Simplifying this equation, we find:
h ≈ 283.8 m
Therefore, the maximum altitude of the rocket is approximately 283.8 meters.
(c) Velocity just before colliding with the Earth:
After the engines fail, the rocket goes into free fall with an acceleration of -9.80 m/s^2. To find the velocity just before it collides with the Earth, we can use the equation:
v = u + gt
We know:
u = 0 m/s (as the rocket is in free fall)
g = -9.80 m/s^2 (acceleration due to gravity)
t = 7.51 s (time when the rocket reaches the maximum altitude)
Substituting these values into the equation, we have:
v = 0 + (-9.80)(7.51)
Simplifying this equation, we find:
v ≈ -73.49 m/s
Therefore, the velocity just before the rocket collides with the Earth is approximately -73.49 m/s (with the negative sign indicating the velocity is directed downward).