An airplane pilot fell 355 m after jumping without his parachute opening. He landed in a snowbank, creating a crater 1.4 m deep, but survived with only minor injuries. Assume that the pilot's mass was 81 kg and his terminal velocity was 50 m/s.
(a) Estimate the work done by the snow in bringing him to rest.
(b) Estimate the average force exerted on him by the snow to stop him.
(c) Estimate the work done on him by air resistance as he fell.
To estimate the work done by the snow in bringing the pilot to a stop, we can use the work-energy principle. The work done by a force is equal to the force applied multiplied by the distance over which it acts. In this case, the force is exerted by the snow, and the distance is the depth of the crater created by the pilot.
(a) Estimate the work done by the snow in bringing him to rest:
We can use the formula W = Fd, where W is work, F is force, and d is distance. In this case, the force exerted by the snow can be assumed to be equal to the average force exerted on the pilot during deceleration. Since we don't know the exact force, we can estimate it by using the pilot's mass and the change in velocity.
The change in velocity can be calculated using the equation vf^2 = vi^2 + 2ad, where vf is the final velocity, vi is the initial velocity (which can be assumed to be the terminal velocity), a is the acceleration (which we need to find), and d is the distance (the depth of the crater). Rearranging the equation, we have:
a = (vf^2 - vi^2) / (2d)
Given that the pilot's terminal velocity is 50 m/s and the depth of the crater is 1.4 m, we can calculate the acceleration:
a = (0 - 50^2) / (2 * -1.4 m) = 625 m^2/s
Now, we can calculate the average force exerted on the pilot during deceleration using the formula F = ma:
F = 81 kg * 625 m^2/s = 50625 N
Finally, we can calculate the work done by the snow using the formula W = Fd:
W = 50625 N * 1.4 m = 70875 J
Therefore, the estimated work done by the snow in bringing the pilot to rest is 70875 Joules.
(b) Estimate the average force exerted on him by the snow to stop him:
We have already calculated the average force exerted on the pilot during deceleration, which is 50625 N.
Therefore, the estimated average force exerted on the pilot by the snow to stop him is 50625 Newtons.
(c) Estimate the work done on him by air resistance as he fell:
The work done by air resistance can be calculated using the formula W = fd, where f is the force of air resistance and d is the distance traveled. To estimate the work done, we need to find the force of air resistance.
Air resistance can be assumed to be equal to the weight of the pilot, as the pilot is falling at their terminal velocity. The weight can be calculated using the formula weight = mass * acceleration due to gravity.
The acceleration due to gravity is approximately 9.8 m/s^2, so the weight of the pilot is:
weight = 81 kg * 9.8 m/s^2 = 794.8 N
Since the pilot falls 355 m, we can calculate the work done by air resistance using the formula W = fd:
W = 794.8 N * 355 m = 282674 J
Therefore, the estimated work done on the pilot by air resistance as he fell is 282674 Joules.