A rocket, initially at rest on the ground, accelerates straight upward from rest with constant acceleration 49.0 . The acceleration period lasts for time 6.00 until the fuel is exhausted. After that, the rocket is in free fall. Find the maximum height reached by the rocket. Ignore air resistance and assume a constant acceleration due to gravity equal to 9.80 .
find the velocity at 6.0 sec
vf=a t
and the height at six seconds:
h=1/2 a t^2, that height becomes the initial height for the gliding phase
the velocity at the top is zero.
vf=0=vi-gt solve for t, where vi is the velociyt at 6.0 sec
so time of flight is 6+t
max height=higluiding+vi*(t)-1/2 g t^2 where t was found above, and vi is the final velocity at t=6 of the propulsion stage
To find the maximum height reached by the rocket, we need to calculate the distance it travels during the acceleration period and then the distance it travels during the free fall period.
1. Calculate the distance traveled during the acceleration period:
During this time, the rocket is undergoing constant acceleration, given by the formula:
d = v_i * t + (1/2) * a * t^2
Where:
- v_i is the initial velocity (0 in this case, as the rocket starts from rest)
- t is the time interval for the acceleration (6.00 s)
- a is the acceleration (49.0 m/s^2)
Plugging in these values, we get:
d = 0 * 6.00 + (1/2) * 49.0 * (6.00)^2
d = 0 + (1/2) * 49.0 * 36.0
d = 882 m
So, during the acceleration period, the rocket travels a distance of 882 meters.
2. Calculate the distance traveled during the free fall period:
After the fuel is exhausted, the rocket is in free fall, experiencing the acceleration due to gravity, g = 9.80 m/s^2. The distance traveled during free fall can be calculated using the formula:
d = v_i * t + (1/2) * g * t^2
Where:
- v_i is the initial velocity (0 in this case, as the rocket is dropped from rest)
- t is the time interval for free fall (unknown)
- g is the acceleration due to gravity (9.80 m/s^2)
We need to find t, the time taken during free fall. Since we know the total time of flight is 6.00 seconds (the time for the acceleration period), we can subtract that from the overall flight time to find the time spent in free fall:
t_freefall = total_time - t_acceleration
t_freefall = 6.00 - 6.00
t_freefall = 0 s
Since t_freefall is 0 seconds, the rocket doesn't spend any time in free fall. This means there is no distance traveled during free fall.
3. Calculate the maximum height reached by the rocket:
The maximum height can be found by adding the distance traveled during the acceleration period to the initial height of the rocket, which is 0 meters (on the ground).
Maximum height = initial height + distance during acceleration
Maximum height = 0 + 882 meters
Maximum height = 882 meters
Therefore, the maximum height reached by the rocket is 882 meters.