A model rocket is launched straight upward with an initial speed of 50m/s. It acccelerates with a constant upward acceleration of 2m/s^2 until its engines stop at an altitude of 150m. (a)What is the maximum height reached by the rocket? (b)How long after lift-off does the rocket reach its maximum height? (c) How long is the rocket in the air?
a) find the velocicy due to the acceleration at the shutoff h=150m. Using that velocity, find the max height (when vfinal=0).
b)how long does it take to reach 150m? Then, how much longer to the top?
To solve this problem, we'll use the equations of motion for uniformly accelerated linear motion. We'll break down the problem into three parts: (a) finding the maximum height reached by the rocket, (b) determining the time it takes to reach the maximum height, and (c) calculating the total time the rocket is in the air.
(a) Finding the maximum height reached by the rocket:
To determine the maximum height, we need to find the time it takes for the rocket's engines to stop. Using the equation v = u + at (where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time), we can find the time it takes for the rocket's engines to stop.
Given:
Initial velocity, u = 50 m/s
Acceleration, a = 2 m/s^2
Final velocity, v = 0 m/s
Rearranging the equation v = u + at, we can solve for time:
0 = 50 + 2t -> -50 = 2t -> t = -50/2 -> t = -25 seconds
Since time cannot be negative, we discard the negative value. Therefore, the rocket's engines stop after 25 seconds.
To find the maximum height, we'll use the equation s = ut + (1/2)at^2 (where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time).
s = (50)(25) + (1/2)(2)(25)^2
s = 1250 + 1/2 * 2 * 625
s = 1250 + 625
s = 1875 m
Therefore, the maximum height reached by the rocket is 1875 meters.
(b) Determining the time it takes to reach the maximum height:
The time it takes to reach the maximum height is half of the total time of flight since the rocket's engines stop at the maximum height.
Total time of flight = 2 * t
Total time of flight = 2 * 25
Total time of flight = 50 seconds
Time to reach maximum height = Total time of flight / 2
Time to reach maximum height = 50 / 2
Time to reach maximum height = 25 seconds
Therefore, the rocket reaches its maximum height 25 seconds after lift-off.
(c) Calculating the total time the rocket is in the air:
The total time the rocket is in the air is the time it takes to reach maximum height plus the time it takes for the rocket to descend from the maximum height back to the ground.
Total time of flight = Time to reach maximum height + Time to descend from maximum height
Since it takes the same amount of time to descend as it does to ascend, the total time of flight is twice the time to reach the maximum height.
Total time of flight = 2 * Time to reach maximum height
Total time of flight = 2 * 25
Total time of flight = 50 seconds
Therefore, the rocket is in the air for 50 seconds.