A helicopter carrying Dr. Evil takes off with a constant upward acceleration of 5.0 m/s2. Secret agent Austin Powers jumps on just as the helicopter lifts off the ground. After the two men struggle for 10.0 s, Powers shuts off the engine and steps out of the helicopter. Assume that the helicopter is in free fall after its engine is shut off, and ignore the effects of air resistance. (a) What is the maximum height above ground reached by the helicopter? (b) Powers deploys a jet pack strapped on his back 7.0 s after leaving the helicopter, and then he has a constant downward acceleration with magnitude 2.0 m/s2. How far is Powers above the ground when the helicopter crashes into the ground?
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To solve this problem, we can break it down into two parts: analyzing the helicopter's motion and analyzing Powers' motion after he leaves the helicopter.
(a) To find the maximum height reached by the helicopter, we need to calculate how long it takes for the helicopter to reach its highest point. We can use the formula:
v = u + at
where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
In this case, the helicopter starts from rest and has an upward acceleration of 5.0 m/s^2. Therefore, the initial velocity is zero, and the final velocity is the maximum height velocity. Substituting these values into the formula, we get:
0 = 0 + 5t
Solving for t, we find t = 0 seconds. This means that the helicopter reaches its highest point instantaneously after takeoff.
To find the height reached, we can use the formula for displacement:
s = ut + 0.5at^2
Since the initial velocity is zero and the time is 10.0 seconds, we have:
s = 0 + 0.5 * 5 * (10)^2
s = 0 + 0.5 * 5 * 100
s = 0 + 0.5 * 500
s = 0 + 250
s = 250 meters
Therefore, the maximum height above the ground reached by the helicopter is 250 meters.
(b) After Powers leaves the helicopter, he deploys a jet pack and experiences a downward acceleration of 2.0 m/s^2. We need to calculate Powers' distance above the ground when the helicopter crashes.
First, we calculate the time it takes for the helicopter to crash. From part (a), we know that the maximum height is reached after 10.0 seconds.
Next, we calculate the distance Powers falls within this time using the kinematic equation:
s = ut + 0.5at^2
where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.
In this case, the initial velocity is zero, the time is 10.0 seconds, and the acceleration is 2.0 m/s^2.
s = 0 + 0.5 * 2 * (10)^2
s = 0 + 0.5 * 2 * 100
s = 0 + 0.5 * 200
s = 0 + 100 meters
Therefore, Powers is 100 meters above the ground when the helicopter crashes into the ground.