A rocket, initially at rest on the ground, accelerates straight upward from rest with constant acceleration 58.8 m/s2 . The acceleration period lasts for time 5.00 s until the fuel is exhausted. After that, the rocket is in free fall.
Max height
Well, let's see. If we consider the rocket accelerating upwards until the fuel is exhausted, we can use the kinematic equation:
y = v0t + (1/2)at^2
Where:
y is the displacement (maximum height),
v0 is the initial velocity,
t is the time, and
a is the acceleration.
Given that the rocket starts from rest, the initial velocity (v0) is 0 m/s. The acceleration (a) is 58.8 m/s^2, and the time (t) is 5.00 seconds.
Plugging these values into the equation, we get:
y = (1/2)(58.8 m/s^2)(5.00 s)^2
y = (1/2)(58.8 m/s^2)(25.00 s^2)
y = 735 m
So, the maximum height the rocket reaches is 735 meters.
Now, if you'll excuse me, I need to go find a ladder to reach that height.
To find the maximum height reached by the rocket, we need to calculate the displacement during the acceleration period and then during the free fall period.
During the acceleration period, we can calculate the displacement using the equation:
s = ut + (1/2)at^2
where:
s = displacement
u = initial velocity (which is zero as the rocket starts from rest)
a = acceleration
t = time
Substituting the given values:
a = 58.8 m/s^2
t = 5.00 s
s = 0 + (1/2)(58.8)(5.00^2)
s = 0 + (1/2)(58.8)(25)
s = 0 + 1470.0
s = 1470.0 m (displacement during acceleration period)
Now, during the free fall period, the rocket is only under the influence of gravity, so the acceleration becomes the acceleration due to gravity (g), which is approximately 9.8 m/s^2.
To find the displacement during free fall, we can use the same equation:
s = ut + (1/2)gt^2
where:
s = displacement
u = initial velocity (which is the final velocity at the end of acceleration period)
g = acceleration due to gravity
t = time
Since the rocket is at rest at the end of the acceleration period, the final velocity is zero.
s = 0 + (1/2)(9.8)(t^2)
s = 0 + (1/2)(9.8)(5.00^2)
s = 0 + (1/2)(9.8)(25)
s = 0 + 122.5
s = 122.5 m (displacement during free fall period)
To find the total maximum height reached by the rocket, we need to add the displacements during the two periods:
Total maximum height = Displacement during acceleration period + Displacement during free fall period
Total maximum height = 1470.0 m + 122.5 m
Total maximum height = 1592.5 m
Therefore, the maximum height reached by the rocket is 1592.5 meters.
To find the maximum height reached by the rocket, we need to consider two phases: the acceleration phase and the free fall phase.
First, let's calculate the maximum height during the acceleration phase.
During this phase, the rocket's initial velocity, u = 0 m/s (since it's at rest), the acceleration, a = 58.8 m/s^2, and the time, t = 5.00 s.
We can use the equation of motion:
h = ut + (1/2)at^2
Substituting the given values:
h = 0 * 5 + (1/2) * 58.8 * (5)^2
h = 0 + (1/2) * 58.8 * 25
h = 0 + 735
h = 735 m
Therefore, during the acceleration phase, the rocket reaches a maximum height of 735 meters.
Now, let's find the maximum height during the free fall phase.
During free fall, the rocket is subject only to the acceleration due to gravity, which is approximately 9.8 m/s^2. The rocket's initial velocity at this point is the final velocity obtained during the acceleration phase.
Given that the acceleration due to gravity is in the opposite direction to the rocket's initial velocity during the acceleration phase, we need to consider it as negative.
Final velocity during the acceleration phase, v = at = 58.8 * 5 = 294 m/s
Using the equation of motion for free fall:
v^2 = u^2 + 2gh
Since the rocket is at rest at maximum height (u = 0), and the acceleration due to gravity, g = -9.8 m/s^2, we can rearrange the equation as follows:
h = (v^2) / (2g)
Substituting the values:
h = (294^2) / (2 * -9.8)
h = 86436 / -19.6
h ≈ -4417.35 meters
Since height cannot be negative, we consider only the magnitude:
h ≈ 4417.35 meters
Therefore, during the free fall phase, the rocket reaches a maximum height of approximately 4417.35 meters.
To find the overall maximum height reached by the rocket, we need to add the heights obtained during both phases:
Overall maximum height ≈ 735 + 4417.35 = 5152.35 meters
Hence, the overall maximum height reached by the rocket is approximately 5152.35 meters.
phase 1
a = 58.8 and t = 5
Hi = (1/2) a t^2 = 29.4*25 = 735 meters
Vi = a t = 58.8 * 5 = 294 m/s
Phase 2
a = -9.81 m/s^2
v = Vi - a t
v = 294 - 9.81 t
v = 0 at top
t = 30 seconds more
h = 735 + 294(30) -4.9 (30^2)
= 735 + 8811 - 4410
= 5136 meters