A model rocket is launched straight upward with an initial speed of 57.0 m/s. It accelerates with a constant upward acceleration of 1.50 m/s2 until its engines stop at an altitude of 190 m.
(a) What is the maximum height reached by the rocket?
(b) How long after liftoff does the rocket reach its maximum height?
(c) How long is the rocket in the air?
figure the velocity at max altitude
vf^2=vi^2-2ad
now, max height:
You know the KE and PE at 190 meters.
ke+PE at 190 =final PE
so now, figure final height max from the final PE at the top.
now time to max height.
Do it in two parts:
a. time up to shutoff: average velocity you know (vfshut+Vi)/2, and distance. Figure height.
b. the coasting part. You know the initial velocity (vfshut), and the final velocity at the top (zero). average velocity then vfshut/2. You know the distance (height in a - 190m).figure time for this leg
add time for both legs.
I got 192.6 meters as my max height, but I don't think that's right. We haven't learned KE and PE yet.
To solve the given problem, we'll use the equations of motion for motion with constant acceleration.
(a) To find the maximum height reached by the rocket, we need to calculate the displacement when the rocket's velocity becomes zero.
We can use the equation: vf^2 = vi^2 + 2ad
Where:
vf = final velocity (0 m/s, as the rocket's engines stop)
vi = initial velocity (57.0 m/s)
a = acceleration (1.50 m/s^2)
d = displacement from the initial point (in this case, the maximum height)
Rearranging the equation, we have:
d = (vf^2 - vi^2) / (2a)
Plugging in the values, we get:
d = (0 - 57.0^2) / (2 * -1.50)
Solving this, we find:
d = -1931.5 / -3 = 643.8 m (rounded to the nearest tenth)
Therefore, the maximum height reached by the rocket is approximately 643.8 m.
(b) To find the time it takes for the rocket to reach its maximum height, let's use the equation with time:
vf = vi + at
Since the final velocity is zero (vf = 0), we can solve for the time (t):
0 = 57.0 + (1.50)(t)
Simplifying:
1.50t = -57.0
t = -57.0 / (1.50)
Therefore, t ≈ 38 seconds (rounded to the nearest whole number).
(c) To find the total time the rocket is in the air, we need to calculate the time it takes to reach the maximum height and double it.
Given that the time to reach the maximum height is approximately 38 seconds, the total time in the air will be 2 times that:
2 * 38 = 76 seconds
Therefore, the rocket is in the air for approximately 76 seconds.