A 60 kg skier leaves the end of a ski-jump ramp with a velocity of 27 m/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 14 m 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 ?

Answer

To solve the problem of how much the mechanical energy of the skier-Earth system is reduced due to air drag, we will follow these steps:

### Step 1: Calculate the Initial Kinetic Energy (KE_initial)
The initial kinetic energy of the skier can be calculated using the formula:
\[ KE_{\text{initial}} = \frac{1}{2} m v_{\text{initial}}^2 \]
where:
- \( m = 60 \, \text{kg} \) (mass of the skier)
- \( v_{\text{initial}} = 27 \, \text{m/s} \) (initial velocity)

Calculating:
\[ KE_{\text{initial}} = \frac{1}{2} \times 60 \times (27)^2 = \frac{1}{2} \times 60 \times 729 = 21870 \, \text{J} \]

### Step 2: Calculate the Final Kinetic Energy (KE_final)
The final kinetic energy of the skier when landing can be calculated using the formula:
\[ KE_{\text{final}} = \frac{1}{2} m v_{\text{final}}^2 \]
where:
- \( v_{\text{final}} = 22 \, \text{m/s} \) (final velocity)

Calculating:
\[ KE_{\text{final}} = \frac{1}{2} \times 60 \times (22)^2 = \frac{1}{2} \times 60 \times 484 = 14520 \, \text{J} \]

### Step 3: Calculate the Work Done by Gravity (W_gravity)
The work done by gravity can be calculated using the formula:
\[ W_{\text{gravity}} = mgh \]
where:
- \( h = 14 \, \text{m} \) (vertical drop)
- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity)

Calculating:
\[ W_{\text{gravity}} = 60 \times 9.81 \times 14 = 8234.4 \, \text{J} \]

### Step 4: Apply the Work-Energy Theorem
According to the work-energy theorem:
\[ W_{\text{gravity}} + W_{\text{air}} = KE_{\text{final}} - KE_{\text{initial}} \]
Rearranging gives us:
\[ W_{\text{air}} = KE_{\text{final}} - KE_{\text{initial}} - W_{\text{gravity}} \]

Substituting the values:
\[ W_{\text{air}} = 14520 - 21870 - 8234.4 \]
Calculating:
\[ W_{\text{air}} = 14520 - 21870 - 8234.4 = -1.6 \times 10^4 \, \text{J} \]

### Step 5: Determine the Loss in Mechanical Energy
The loss in mechanical energy of the skier-Earth system due to air drag is equal to the negative work done by air:
\[ \text{Loss in Mechanical Energy} = -W_{\text{air}} = 1.6 \times 10^4 \, \text{J} \]

### Final Answer
The mechanical energy of the skier-Earth system is reduced by \( 1.6 \times 10^4 \, \text{J} \) due to air drag.

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