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

To determine the reduction in mechanical energy of the skier-Earth system due to air drag, we need to analyze both the initial and final mechanical energy of the skier during the jump. Mechanical energy consists of kinetic energy (KE) and potential energy (PE). Let's break this down step by step.

Initial Mechanical Energy Calculation

The skier's initial mechanical energy when leaving the ramp can be calculated using the following formulas:

• Kinetic Energy (KE): KE = (1/2)mv²
• Potential Energy (PE): PE = mgh

Here, m is the mass of the skier (60 kg), v is the initial velocity (24 m/s), g is the acceleration due to gravity (approximately 9.81 m/s²), and h is the height from which the skier jumps. To find h, we can use the vertical component of the initial velocity.

Finding the Initial Height

The vertical component of the initial velocity (v_y) can be calculated as:

v_y = v * sin(θ) = 24 m/s * sin(25°) ≈ 10.1 m/s

Using the kinematic equation for vertical motion, we can find the maximum height (h) reached by the skier:

h = (v_y²) / (2g) = (10.1 m/s)² / (2 * 9.81 m/s²) ≈ 5.1 m

Now, we can calculate the initial potential energy:

PE_initial = mgh = 60 kg * 9.81 m/s² * 5.1 m ≈ 2994.6 J

Next, we calculate the initial kinetic energy:

KE_initial = (1/2)mv² = (1/2) * 60 kg * (24 m/s)² = 17280 J

Thus, the total initial mechanical energy (E_initial) is:

E_initial = KE_initial + PE_initial = 17280 J + 2994.6 J ≈ 20274.6 J

Final Mechanical Energy Calculation

Upon landing, the skier's speed is 22 m/s, and they are 14 m below the ramp. Therefore, we can calculate the final potential energy and kinetic energy:

Final height (h_final) = -14 m (since it's below the ramp)

PE_final = mgh_final = 60 kg * 9.81 m/s² * (-14 m) ≈ -8234.4 J

Now, we calculate the final kinetic energy:

KE_final = (1/2)mv² = (1/2) * 60 kg * (22 m/s)² = 14520 J

Thus, the total final mechanical energy (E_final) is:

E_final = KE_final + PE_final = 14520 J - 8234.4 J ≈ 6285.6 J

Energy Loss Due to Air Drag

Now, we can find the reduction in mechanical energy due to air drag by subtracting the final mechanical energy from the initial mechanical energy:

Energy loss = E_initial - E_final = 20274.6 J - 6285.6 J ≈ 13989 J

In conclusion, the mechanical energy of the skier-Earth system is reduced by approximately 13989 Joules due to air drag during the jump.