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.
Vo = a*t = 58.8 * 5 = 294 m/s. = Velocity at the beginning of free fall period.
V^2 = Vo^2 + 2g*h = 0 @ max ht.
(294)^2 - 19.6h = 0.
h = ?.
ho = 0.5a*t^2 = .5*58.8*5^2 = 735 m. = Ht. @ the beginning of free fall period.
ho + h = Ht. above gnd.
To solve this problem, we can break it down into a few steps:
Step 1: Find the time it takes for the rocket to reach its maximum speed.
Given:
Initial velocity (u) = 0 m/s
Acceleration (a) = 58.8 m/s^2
Time (t) = 5.00 s
Using the equation of motion (v = u + at), we can find the final velocity (v) at the end of the acceleration period:
v = u + at
v = 0 + 58.8 * 5
v = 294 m/s
Step 2: Find the distance covered during the acceleration period.
To find the distance covered during the acceleration period, we can use the equation of motion (s = ut + 0.5at^2):
s = ut + 0.5at^2
s = 0 + 0.5 * 58.8 * (5)^2
s = 735 m
So, during the acceleration period, the rocket covers a distance of 735 meters.
Step 3: Find the time it takes for the rocket to reach the ground after fuel is exhausted.
Since the rocket is now in free fall, the time it takes to reach the ground can be found using the equation s = ut + 0.5gt^2, where g represents the acceleration due to gravity.
Given:
Initial velocity (u) = 294 m/s (the maximum speed reached during acceleration)
Acceleration due to gravity (g) = 9.8 m/s^2 (assuming no air resistance)
Distance (s) = ?
To find the distance, we need to calculate how much time it takes for the rocket to reach the ground. At this moment, the initial velocity becomes 0 (u = 0) since the rocket is starting its fall from rest.
s = ut + 0.5gt^2
0 = 294t + 0.5 * 9.8 * t^2
0 = 294t + 4.9t^2
Solving this quadratic equation, we find t = 30.0 s or t = -10.0 s. Since time cannot be negative, the rocket takes 30.0 seconds to reach the ground.
Step 4: Find the distance covered during free fall.
To find the distance covered during free fall, we can use the equation of motion (s = ut + 0.5gt^2):
s = ut + 0.5gt^2
s = 0 + 0.5 * 9.8 * (30)^2
s = 4410 m
So, during the free fall, the rocket covers a distance of 4410 meters.
In summary, the rocket covers:
- 735 meters during acceleration
- 4410 meters during free fall.
To find the maximum height reached by the rocket, we need to analyze its motion in two parts: the upward acceleration phase and the free fall phase.
1. Upward Acceleration Phase:
During this phase, the rocket starts from rest and accelerates upwards at a constant rate until its fuel is exhausted. We can use the kinematic equation for motion with constant acceleration to find the height reached during this phase.
The equation we can use is:
h = v₀t + (1/2)at²
Where:
- h is the height
- v₀ is the initial velocity (which is zero in this case)
- t is the time
- a is the acceleration
In this case, the initial velocity is zero, the acceleration is 58.8 m/s², and the time is 5.00 s. Plugging these values into the equation, we can find the height reached during the acceleration phase.
2. Free Fall Phase:
After the fuel is exhausted, the rocket is no longer accelerating and enters free fall. In this phase, the rocket will be under the influence of gravity alone, which causes it to fall downward. The height reached during the free fall phase is given by the equation:
h = (1/2)gt²
Where:
- h is the height
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- t is the time
Since the rocket is now falling, the time will be the total time elapsed from the start of the upward acceleration phase. In this case, the time is 5.00 seconds.
Finally, to find the maximum height reached by the rocket, we need to add the heights from both phases. The total height is the height reached during the acceleration phase plus the height reached during the free fall phase.
I hope this explanation helps! If you have any further questions, feel free to ask.