A model rocket is launched straight upward
with an initial speed of 45.4 m/s. It accelerates with a constant upward acceleration of
2.98 m/s
2
until its engines stop at an altitude
of 150 m.
What is the maximum height reached by
the rocket? The acceleration of gravity is
9.81 m/s
2
.
Answer in units of m
Vf^2 = Vo^2 + 2g*d,
Vf^2 = (45.4)^2 + 19.6*150 = 5001.16,
Vf = 70.7m/s = Vo for free fall phase.
h = ho + (Vf^2 - Vo^2) / 2g,
h = 150 + (0 - (70.7)^2) / -19.6=405m.
When does the rocket reach maximum height?
Answer in units of s
How long is the rocket in the air?
Answer in units of s
Correction to your 10-2-11,11:32am post:
The acceleration for the 1st phase should be 2.98m/s :
Vf^2 = Vo^2 + 2a*d,
Vf^2 = (45.4)^2 + 5.96*150 = 2955.16,
Vf = 54.36m/s. = Vo for free-fall phase
h = ho + (Vf^2 - Vo^2) / 2g,
h=150 + (0 - (54.36)^2) / -19.6 = 301m.
t1 = (Vf - Vo) / a,
t1 = (54.36 - 45.4) / 2.98 = 3.0s. to
reach 150m.
t2 = (0 - 54.36) / -9.8 = 5.55s To go
from 150m to max. height(301m).
t(up) = t1 + t2 = 3 + 5.55 = 8.55s
d = V0*t + 0.5g*t^2 = 301m,
0 + 4.9t^2 = 301,
t^2 = 61.4,
t(dn) = 7.84s.
T = t(up) + t(dn) = 8.55 + 7.84 = 16.4s = Time in flight.
To find the maximum height reached by the rocket, we need to apply the kinematic equations of motion.
First, we need to find the time it takes for the engines to stop. We can use the equation:
vf = vi + at
where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time taken.
Initially, the rocket is moving upward with an initial velocity of 45.4 m/s, and it accelerates upwards with a constant acceleration of 2.98 m/s^2. We want to find the time when the velocity becomes zero (vf = 0).
0 = 45.4 + (-2.98)t
Rearranging the equation, we have:
2.98t = 45.4
t = 45.4 / 2.98
t ≈ 15.216 seconds
Therefore, it takes approximately 15.216 seconds for the rocket engines to stop.
Now, we can find the maximum height reached by the rocket using the equation:
h = vi * t + (1/2) * a * t^2
where h is the height, vi is the initial velocity, t is the time, and a is the acceleration.
Plugging in the values:
h = 45.4 * 15.216 + (1/2) * (-9.81) * (15.216)^2
h ≈ 345.7792 + (-74.3687)
h ≈ 271.4105 meters
Therefore, the maximum height reached by the rocket is approximately 271.4105 meters.