To solve parts (a) and (b), we need to use the principles of work and energy.
(a) The rate at which the gravitational force does work on the skier can be calculated by finding the component of the gravitational force parallel to the motion of the skier when he leaves the ramp.
The gravitational force can be calculated using the formula:
F_gravity = m * g
where m is the mass of the skier and g is the acceleration due to gravity.
The component of the gravitational force parallel to the motion of the skier can be calculated using:
F_parallel = F_gravity * sin(θ)
where θ is the angle between the ramp and the horizontal direction.
The rate at which the gravitational force does work on the skier can be found by multiplying the component of the force parallel to the motion by the velocity of the skier:
Rate of work = F_parallel * v
Now, let's calculate the values.
Given:
Mass of the skier, m = 72 kg
Velocity of the skier, v = 24 m/s
Angle with the horizontal, θ = 25°
Acceleration due to gravity, g ≈ 9.8 m/s²
Calculations:
F_gravity = m * g
= 72 kg * 9.8 m/s²
= 705.6 N
F_parallel = F_gravity * sin(θ)
= 705.6 N * sin(25°)
≈ 299.2 N
Rate of work = F_parallel * v
= 299.2 N * 24 m/s
≈ 7170.8 J/s
≈ 7170.8 Watts
Hence, the rate at which the gravitational force does work on the skier when he leaves the ramp is approximately 7170.8 Watts.
(b) To find the total speed with which the skier would have landed if there were no air drag, we can use the principle of conservation of mechanical energy.
The initial mechanical energy of the skier is the sum of his kinetic energy and potential energy when he leaves the ramp.
The final mechanical energy of the skier is the sum of his kinetic energy and potential energy when he returns to the ground.
Assuming there is no loss of mechanical energy due to air drag, we can equate the initial mechanical energy to the final mechanical energy.
Initial mechanical energy = Final mechanical energy
(1/2) * m * v_initial^2 + m * g * h = (1/2) * m * v_final^2 + m * g * h_final
where v_initial is the initial velocity of the skier, h is the height of the ramp, v_final is the final velocity of the skier, and h_final is the height of the ground.
Given:
Mass of the skier, m = 72 kg
Initial velocity of the skier, v_initial = 24 m/s
Height of the ramp, h = 0 (since the skier starts at the end of the ramp)
Final velocity of the skier, v_final = 22 m/s
Height of the ground, h_final = -14 m (negative since it is below the ramp)
Calculations:
(1/2) * m * v_initial^2 + m * g * h = (1/2) * m * v_final^2 + m * g * h_final
(1/2) * 72 kg * (24 m/s)^2 + 72 kg * 9.8 m/s^2 * 0 = (1/2) * 72 kg * (22 m/s)^2 + 72 kg * 9.8 m/s^2 * (-14 m)
20736 J + 0 = 17424 J + (-14256 J)
20736 J = 3168 J
This is not a valid equation, and it indicates that the assumption of no air drag is not valid. In reality, air drag causes some energy loss, so the skier will not have the same total speed when landing.
Therefore, we cannot determine the total speed with which the skier would have landed if there were no air drag (part b).
I hope this helps! Let me know if you have any further questions.