A rocket is launched vertically with an acceleration of 20m/s^2. After two minutes its engines fail. Calculate the max height reached by the rocket and its total time in the air
after 120 seconds,
v = at = 20*120 = 2400 m/s
h = 1/2 at^2 = 1/2 * 20 * 120^2 = 144000 m
From that point, it is just a ballistic trajectory, and the height is just
h(t) = 144000 + 2400t - 4.9t^2
Now just find the vertex of that parabola, in the usual way.
Then solve for t when h=0
That value for t will be after the initial 120 seconds ...
To find the maximum height reached by the rocket and its total time in the air, we need to use the kinematic equations for uniformly accelerated motion.
Let's break the problem down into steps:
Step 1: Find the time taken for the engines to fail.
Given:
Initial acceleration (a) = 20 m/s^2
Time (t) = 2 minutes = 2 * 60 = 120 seconds (since there are 60 seconds in a minute)
We know that the rocket is launched vertically, so the initial velocity is 0 m/s.
Using the formula: 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 solve for t:
0 (final velocity) = 0 (initial velocity) + 20 (acceleration) * t
0 = 20t
So, t = 0 seconds.
Therefore, the engines fail immediately upon launch.
Step 2: Find the maximum height reached by the rocket.
Since the acceleration is constant and the rocket starts from rest, we can use the equation: s = ut + 1/2 at^2
Given:
Initial velocity (u) = 0 m/s
Acceleration (a) = 20 m/s^2
Time (t) = 120 seconds
Plugging in these values, we get:
s = 0 * 120 + 1/2 * 20 * (120)^2
s = 0 + 1/2 * 20 * 14400
s = 0 + 144000
s = 144,000 meters
Therefore, the maximum height reached by the rocket is 144,000 meters.
Step 3: Find the total time in the air
Since the engines failed immediately upon launch, we can calculate the total time the rocket is in the air by adding the time taken for the engines to fail (which is 0 seconds) to the time it took to reach the maximum height.
Total time in the air = 0 seconds + 120 seconds
Total time in the air = 120 seconds
Therefore, the rocket is in the air for a total of 120 seconds.