A rocket rises vertically, from rest, with an acceleration of 3.2 m/s2 until it runs out of fuel at an altitude of 730m . After this point, its acceleration is that of gravity, downward.whats the maximum altitude does the rocket reach?
To find the maximum altitude reached by the rocket, we need to determine the time it takes for the rocket to reach its highest point.
First, we can find the time it takes for the rocket to run out of fuel using the equation of motion:
vf = vi + at
where vf is the final velocity, vi is the initial velocity, a is the acceleration, and t is the time.
In this case, the rocket starts from rest (vi = 0) and has an acceleration of 3.2 m/s² until it runs out of fuel. Let's assume the time taken for the fuel to run out is "t1".
So, using the equation above, we get:
0 + (3.2 * t1) = vf
Next, we can use the equation for displacement to find the time it takes for the rocket to reach its highest point:
d = vit + (1/2)at²
where d is the displacement (altitude), vi is the initial velocity, a is the acceleration, and t is the time.
At the highest point, the rocket momentarily comes to a stop, so the final velocity is 0. Therefore, we have:
0 = vi + at2/2
As the acceleration after the fuel runs out is the acceleration due to gravity (g ≈ 9.8 m/s²), let's assume the time taken from the rocket running out of fuel to the highest point is "t2".
Now, we can calculate the maximum altitude using the equations derived:
1. For the time taken for the fuel to run out:
0 + (3.2 * t1) = 0
t1 = 0
2. For the time taken from the rocket running out of fuel to the highest point:
0 = 0 + (9.8 * t2²)/2
t2² = 0
t2 = 0
From the values obtained, we see that both t1 and t2 are equal to 0. This implies that the rocket reaches its maximum altitude as soon as the fuel runs out, without any additional time to ascend.
Therefore, the maximum altitude reached by the rocket is 730 meters.