To find the maximum height reached by the rocket, we need to determine the time it takes for the rocket's engines to stop. We can then use this time to calculate the displacement at that point.
First, let's find the time it takes for the rocket's engines to stop using the equation of motion:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Given:
Initial velocity, u = 53.0 m/s
Acceleration, a = 3.21 m/s^2
Final velocity, v = 0 m/s (since the engines stop at maximum height)
Rearranging the equation, we have:
t = (v - u) / a
t = (0 - 53.0) / 3.21 = -53.0 / 3.21 ≈ -16.48 s (Taking the negative value indicates that the rocket is moving opposite to the positive direction of displacement)
Since time cannot be negative, we ignore the negative sign and take the absolute value:
t = 16.48 s
Now, we can calculate the maximum height reached by the rocket using the equation of motion:
s = ut + (1/2)at^2
where s is the displacement or height.
Given:
Initial velocity, u = 53.0 m/s
Acceleration, a = -9.81 m/s^2 (negative because the rocket is moving upward against gravity)
Time, t = 16.48 s
Substituting the values into the equation:
s = (53.0)(16.48) + (1/2)(-9.81)(16.48)^2
s = 873.44 + (- 1320.77)
s ≈ -447.33 m
Again, since displacement cannot be negative for height, we ignore the negative sign:
s ≈ 447.33 m
Therefore, the maximum height reached by the rocket is approximately 447.33 meters.