Sam, whose mass is 73  kg  , stands at the top of a 11- m -high, 120- m -long snow-covered slope. His skis have a coefficient of kinetic friction on snow of 0.07. If he uses his poles to get started, then glides down, what is his speed at the bottom?

To determine Sam's speed at the bottom of the slope, we can apply the principle of conservation of energy, factoring in the work done by friction. Here's how we can solve this step by step:

### Step 1: Identify the forces involved
1. **Gravitational potential energy (PE):** This is the energy Sam has at the top of the slope due to his height. It is given by:
   \[
   PE = mgh
   \]
   where:
   - \( m = 73 \, \text{kg} \) (Sam's mass),
   - \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity),
   - \( h = 11 \, \text{m} \) (height of the slope).

2. **Kinetic energy (KE):** This is the energy Sam has due to his velocity as he glides down the slope. At the bottom, his potential energy is fully converted into kinetic energy, minus the energy lost due to friction.

3. **Work done by friction (W_{\text{friction}}):** Friction acts along the slope, doing negative work and reducing Sam's kinetic energy. The work done by friction is:
   \[
   W_{\text{friction}} = f_k \cdot d
   \]
   where:
   - \( f_k = \mu_k \cdot N \) is the kinetic friction force,
   - \( \mu_k = 0.07 \) is the coefficient of kinetic friction,
   - \( N = mg \cdot \cos(\theta) \) is the normal force,
   - \( d = 120 \, \text{m} \) is the length of the slope,
   - \( \theta \) is the angle of the slope.

### Step 2: Calculate the angle of the slope (\( \theta \))
We know the slope's height (\( h = 11 \, \text{m} \)) and length (\( d = 120 \, \text{m} \)), so we can calculate the angle \( \theta \) using trigonometry:
\[
\sin(\theta) = \frac{h}{d}
\]

### Step 3: Apply conservation of energy
At the top, Sam has potential energy:
\[
PE_{\text{top}} = mgh
\]
At the bottom, Sam's kinetic energy will be:
\[
KE_{\text{bottom}} = \frac{1}{2}mv^2
\]
The total mechanical energy lost due to friction is equal to the work done by friction:
\[
W_{\text{friction}} = f_k \cdot d = \mu_k \cdot mg \cdot \cos(\theta) \cdot d
\]
Thus, the energy balance equation is:
\[
mgh - \mu_k \cdot mg \cdot \cos(\theta) \cdot d = \frac{1}{2}mv^2
\]
Solve this equation for \( v \), Sam's speed at the bottom.

### Step 4: Calculate

I'll calculate this for you.Sam's speed at the bottom of the slope is approximately **7.19 m/s**.

Would you like a more detailed breakdown of any part of this solution? Here are 5 related questions to explore further:

1. How does increasing the slope length affect the final speed?
2. What if the coefficient of friction was zero (no friction at all)?
3. How would Sam's speed change if his mass was different?
4. What effect would a steeper slope (larger height) have on the final speed?
5. How is the normal force calculated for inclined planes?

**Tip:** In problems involving energy and friction, always check whether non-conservative forces like friction are doing significant work, as they reduce the mechanical energy available for motion.

Sam, whose mass is 73 kg, stands at the top of a 11-m-high, 120-m-long snow-covered slope. His skis have a coefficient of kinetic friction on snow of 0.07. If he uses his poles to get started, then glides down, what is his speed at the bottom?

Calculate Sam's speed at the bottom of a 120-meter snow-covered slope with a 0.07 coefficient of kinetic friction. Explore conservation of energy, work done by friction, and slope geometry. This detailed solution applies trigonometry and physics principles, suitable for Grades 10-12.

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