A model rocket is launched straight upward
with an initial speed of 53.0 m/s. It acceler-
ates with a constant upward acceleration of
3.21 m/s2 until its engines stop at an altitude
of 230 m.
What is the maximum height reached by
the rocket? The acceleration of gravity is
9.81 m/s2.
Answer in units of
two phases, accelerating up and coasting up
acceleration phase:
V = Vi + a t
h = Vi t + (1/2) a t^2
230 = 53 t + 1.605 t^2
solve this quadratic equation for t
then v at 230 m = 53 + 3.21 t
Now phase 2, coasting up from 230 m with new Vi and acceleration = -9.8 m/s
Vi = the v we just got from phase one.
final v at top = 0
so
0 = Vi - 9.8 t
solve for t
then
h = 230 + Vi t - 4.9 t^2
Vi =
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.