A rocket blasts off with acceleration of 5 m/s squared. After 1 min it runs out of fuel. What is the max height and total time in the air?
To find the maximum height and total time in the air, we need to break down the problem into two parts: the upward motion while the rocket has fuel and the downward motion after the fuel is depleted.
First, let's calculate the time it takes for the rocket to run out of fuel. We know that the acceleration is 5 m/s² and the time is 1 minute, which is equivalent to 60 seconds. We can use the equation of motion:
v = u + at
where
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the rocket starts from rest (u = 0), we can simplify the equation to:
v = at
Plugging in the given values:
v = 5 m/s² * 60 s = 300 m/s
The rocket's velocity when it runs out of fuel is 300 m/s.
Now, let's find the time it takes for the rocket to reach the maximum height. This can be done by using the kinematic equation:
v² = u² + 2as
where
s = displacement
Since the rocket's final velocity is 0 m/s at maximum height, the equation becomes:
0 = v² + 2as
Rearranging the equation:
s = -v² / (2a)
Plugging in the values:
s = -(300 m/s)² / (2 * -5 m/s²) = 9000 m
Thus, the maximum height reached by the rocket is 9000 meters.
The total time in the air can be calculated by adding the time it takes to reach the maximum height and the time it takes for the rocket to run out of fuel. Since the upward and downward motions are symmetrical, the total time in the air is twice the time it takes to reach the maximum height:
Total time = 2 * time to reach maximum height
The time to reach maximum height can be calculated using the equation of motion:
v = u + at
where
v = final velocity
u = initial velocity
a = acceleration
t = time
Since the final velocity is 0 m/s (at maximum height), we can substitute into the equation:
0 = u + (-5 m/s²) * t
Rearranging the equation:
t = u / 5
The initial velocity (u) of the rocket at maximum height is the same as the velocity when it runs out of fuel, so we can use the value we calculated earlier: u = 300 m/s.
t = 300 m/s / 5 m/s² = 60 seconds
Thus, the total time in the air is:
Total time = 2 * 60 seconds = 120 seconds.
Therefore, the maximum height reached by the rocket is 9000 meters, and it stays in the air for a total of 120 seconds.