A model rocket is launched straight upward with an initial speed of 40.0 m/s. It accelerates with a constant upward acceleration of 2.50 m/s2 until its engines stop at an altitude of 120 m.
To solve this problem, we can use kinematic equations of motion to find the time, maximum height reached, and the velocity when the rocket's engines stop.
First, let's identify the given values:
Initial velocity (u) = 40.0 m/s (upward)
Acceleration (a) = 2.50 m/s^2 (upward)
Final displacement (s) = 120 m (upward)
Now, using the second equation of motion:
s = ut + (1/2)at^2
We want to find the time (t) it takes for the rocket to reach an altitude of 120 m. Rearranging the equation, we get:
t = √((2s)/a)
Substituting the given values, we have:
t = √((2 * 120) / 2.5)
t = √(240 / 2.5)
t = √96
t ≈ 9.80 seconds
So, it takes approximately 9.80 seconds for the rocket's engines to stop.
Now, let's find the maximum height reached by the rocket. We can use the first equation of motion:
v = u + at
At maximum height, the velocity (v) of the rocket is 0 m/s (since it stops moving). Rearranging the equation, we get:
t = (v - u) / a
Substituting the given values:
t = (0 - 40) / (-2.5)
t = (-40) / (-2.5)
t = 16 seconds
Now, we know it takes 16 seconds for the rocket to reach maximum height. To find the maximum height (h), we can use the equation:
h = ut + (1/2)at^2
Substituting the values:
h = (40 * 16) + (0.5 * (-2.5) * 16^2)
h = 640 - 0.5 * 10 * 256
h = 640 - 1280
h = -640
The maximum height reached by the rocket is -640 meters. Since the height can't be negative, we know there may have been some mistake in our calculations or assumptions. Please double-check the given information and calculations to determine the correct maximum height.