A rocket, initially at rest on the ground, accelerates straight upward from rest with constant acceleration 49.0 . The acceleration period lasts for time 10.0 until the fuel is exhausted. After that, the rocket is in free fall.
What's the max height? Ignore air resistance and assume a constant acceleration due to gravity equal to 9.80
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To find the maximum height reached by the rocket, we need to break down the problem into two parts: the ascending phase with constant acceleration and the descending phase in free fall.
1. Ascending Phase:
During the ascending phase, the rocket is accelerating upwards with a constant acceleration of 49.0 m/s² until the fuel is exhausted.
Using the kinematic equation, we can determine the displacement of the rocket during this time:
s = ut + (1/2)at²
Where:
s = displacement (unknown)
u = initial velocity (0 m/s, as the rocket starts from rest)
a = acceleration (49.0 m/s²)
t = time (10.0 s, as given)
Plugging in the values, we get:
s = 0 + (1/2)(49.0)(10.0)²
s = 1/2 * 49.0 * 100.0
s = 2450.0 meters
So, during the ascending phase, the rocket reaches a height of 2450.0 meters.
2. Descending Phase (Free Fall):
After the fuel is exhausted, the rocket enters the descending phase, where it is in free fall. During free fall, the rocket is subjected to the acceleration due to gravity, which is 9.80 m/s² in this case.
To find the time it takes for the rocket to reach its maximum height during free fall, we can use the following equation of motion:
v = u + at
Where:
v = final velocity (0 m/s, as the rocket comes to rest at its maximum height)
u = initial velocity (unknown)
a = acceleration (-9.80 m/s², as the rocket is moving in the opposite direction of upward motion)
t = time (unknown)
Plugging in the values, we have:
0 = u + (-9.80)(t)
u = 9.80t
Now, we know that the total time for the ascending phase was 10.0 seconds. Therefore, the time for the descending phase will also be 10.0 seconds.
Substituting the time (t = 10.0 s) into the equation, we get:
u = 9.80 * 10.0
u = 98.0 m/s
So, the initial velocity of the rocket during the descending phase is 98.0 m/s.
To find the maximum height, we can again use the kinematic equation:
s = ut + (1/2)at²
Where:
s = displacement (unknown, the maximum height we want to find)
u = initial velocity (98.0 m/s)
a = acceleration (-9.80 m/s²)
t = time (10.0 s)
Plugging in the values, we get:
s = (98.0 * 10.0) + (1/2)(-9.80)(10.0)²
s = 980.0 - (1/2)(9.80)(100.0)
s = 980.0 - 490.0
s = 490.0 meters
Therefore, the maximum height reached by the rocket is 490.0 meters.
Note: This calculation assumes that the ground level is taken as the reference point (i.e., the displacement is measured from the ground level).