A model rocket is launched straight upward with an initial speed of 58.0 m/s. It accelerates with a constant upward acceleration of 1.50 m/s2 until its engines stop at an altitude of 120 m.
(a) What can you say about the motion of the rocket after its engines stop?
(b) What is the maximum height reached by the rocket?
(c) How long after liftoff does the rocket reach its maximum height?
(d) How long is the rocket in the air?
(a) At rocket cutoff, it changes from accelerating at 1.5 m/s^2 to decelerating at rate -g = -9.8 m/s^2.
(b) At 120 m cutoff, the rocket's velocity V is given by
V^2 - 58^2 = 2 a X = 360 m^2/s^2
V = 61.02 m/s
Maximum height occurs t = 61.02/9.8 = 6.23 s later. Additional height above 120 m is
(V/2)*t = 190.1 m higher, or 310.1 m
(c) already answered. See t.
(d) During acceleration, the time the rocket was burning was
120 m/(Vaverage) = 2*120/(58 +61)
= 2.02 s.
Add that to 6.23 s for the time to reach maximum height. Then add the time it takes to fall from there.
To answer these questions, we can use the equations of motion for uniformly accelerated motion. The important equation in this case is:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
(a) After the rocket's engines stop, it will continue to move upward due to its initial upward velocity, but its acceleration will now be due to gravity pulling it downward. Therefore, the rocket's motion after the engines stop will be in the form of free fall.
(b) To find the maximum height reached by the rocket, we need to find the displacement when the rocket's velocity reaches zero. At that point, the rocket will be at its maximum height. We can use the equation mentioned above, setting v = 0, u = 58.0 m/s, and a = -9.8 m/s^2 (acceleration due to gravity):
0^2 = (58.0 m/s)^2 + 2(-9.8 m/s^2)s
Simplifying this equation, we get:
-2(-9.8 m/s^2)s = (58.0 m/s)^2
Solving for s, we find:
s = [(58.0 m/s)^2] / (2(-9.8 m/s^2))
Calculating this, we get:
s = 169.43 m
Therefore, the maximum height reached by the rocket is 169.43 m.
(c) To find the time it takes for the rocket to reach its maximum height, we can use the equation:
v = u + at
where t is the time taken and v is the final velocity (which is 0 when the rocket reaches its maximum height). Rearranging the equation, we have:
t = (v - u) / a
Substituting the given values, we get:
t = (0 - 58.0 m/s) / (-9.8 m/s^2)
Calculating this, we find:
t = 5.92 s
Therefore, the rocket reaches its maximum height 5.92 seconds after liftoff.
(d) The total time the rocket is in the air can be calculated by doubling the time it takes to reach the maximum height (as it takes the same amount of time to come back down). Thus, the total time in the air is:
2 * 5.92 s = 11.84 s
Therefore, the rocket is in the air for 11.84 seconds.