A 60 Kg skier leaves the end of a ski-jump ramp with a velocity of 24m/s directed 25° above the horizontal. Suppose that as a result of air drag the skier returns to the ground with a speed of 22 m/s, landing 14m vertically below the end of the ramp. From the launch to the return to the ground, by how much is the mechanical energy of the skier-Earth system reduced because of air drag
Add the kinetic energy loss,
(1/2)M[(24^2 - 22^2] to the potential enenrgy loss,
M g H. (H = 14 m)
The launch angle doesn't matter.
To determine the reduction in mechanical energy of the skier-Earth system due to air drag, we need to calculate the initial and final mechanical energies and then find their difference.
The initial mechanical energy of the skier-Earth system is given by the sum of the skier's kinetic energy and potential energy:
Initial mechanical energy = Kinetic energy + Potential energy
The kinetic energy can be calculated using the formula:
Kinetic energy = (1/2) * mass * velocity^2
Where
mass = 60 kg (mass of the skier)
velocity = 24 m/s (launch velocity)
The potential energy at the end of the ramp can be calculated using the formula:
Potential energy = mass * g * height
Where
mass = 60 kg (mass of the skier)
g = acceleration due to gravity ≈ 9.8 m/s^2
height = vertical distance above the ground = 0m
Hence, the initial mechanical energy is:
Initial mechanical energy = (1/2) * 60 kg * (24 m/s)^2 + 60 kg * 9.8 m/s^2 * 0 m
= 20,160 J
The final mechanical energy of the skier-Earth system is the sum of the skier's kinetic energy and potential energy at the landing point.
The kinetic energy at the landing can be calculated using the formula mentioned earlier by substituting the velocity at the landing point, which is 22 m/s.
The potential energy at the landing point can be calculated using the formula mentioned earlier by substituting the height, which is 14m below the end of the ramp.
Hence, the final mechanical energy is:
Final mechanical energy = (1/2) * 60 kg * (22 m/s)^2 + 60 kg * 9.8 m/s^2 * (-14 m)
= 14,388 J
Finally, we can determine the reduction in mechanical energy:
Reduction in mechanical energy = Initial mechanical energy - Final mechanical energy
= 20,160 J - 14,388 J
= 5,772 J
Therefore, the mechanical energy of the skier-Earth system is reduced by 5,772 Joules due to air drag.