A rocket, initially at rest on the ground, accelerates straight upward with a constant acceleration of 34.3 . The rocket accelerates for a period of 8.00 before exhausting its fuel. The rocket continues its ascent until its motion is halted by gravity. The rocket then enters free fall.Part A: Find the maximum height, , reached by the rocket. Ignore air resistance and assume a constant acceleration due to gravity of 9.810 ..
h1 = Vo*t + 0.5*at^2,
h1 = 0 + 17.15*8^2 = 1097.6m.
Vf^2 = Vo^2 + 2ad,
Vf^2 = 0 + 68.6*1097.6 = 75,295.4,
Vf = 274.4m/s = Vo for 2nd phase.
h2 = (Vf^2 - Vo^2) / 2g,
h2 = (0 - (274.4)^2) / -19.6 = 3842m.
h = h1 + h2 = 1097.6 + 3842 = 4940m.
To find the maximum height reached by the rocket, we need to determine the time it takes for the rocket to reach its highest point.
We can start by analyzing the rocket's upward motion during the period of acceleration. Given that the rocket initially starts from rest and has a constant acceleration of 34.3 m/s^2, we can use the kinematic equation:
vf = vi + at
where:
vf = final velocity
vi = initial velocity
a = acceleration
t = time
Initially, the rocket's velocity is 0 m/s (since it starts from rest), and the acceleration is 34.3 m/s^2. We want to find the time it takes for the acceleration period only, so we can rewrite the equation:
vf = 0 + (34.3 m/s^2)(8.00 s)
Simplifying, we find that the final velocity after 8.00 seconds of acceleration is:
vf = 274.4 m/s
Now, we can find the time it takes for the rocket to reach its highest point by using the equation:
vf = vi + at
where:
vi = initial velocity (which is the final velocity after the acceleration period)
a = acceleration due to gravity (which is -9.810 m/s^2, considering upward motion as positive and downward motion as negative)
t = time to highest point
Plugging in the values we know:
0 = 274.4 m/s + (-9.810 m/s^2)t
Solving for t gives us:
t = -274.4 m/s / -9.810 m/s^2
t ≈ 28 s
Therefore, it takes approximately 28 seconds for the rocket to reach its highest point.
To find the maximum height, we can use the equation:
yt = yi + vit + (1/2)at^2
where:
yt = height at time t
yi = initial height (which is 0 m since the rocket starts from the ground)
vi = initial velocity (which is the final velocity after the acceleration period)
a = acceleration due to gravity
t = time to highest point
Substituting the values:
yt = 0 + (274.4 m/s)(28 s) + (1/2)(-9.810 m/s^2)(28 s)^2
Calculating this equation gives us:
yt ≈ 3833 m
The maximum height reached by the rocket is approximately 3833 meters.