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We are given the following information from the problem:

- The skier's initial speed: \( v_0 = 16.6 \, \text{m/s} \)
- The angle of the slope below the horizontal: \( \theta = 31.5^\circ \)
- The vertical height difference between the starting point and the opposite side of the gorge: \( h = 17.1 \, \text{m} \)

We need to find the maximum width of the gorge \( w \), which can be done by analyzing the projectile motion of the skier.

### Steps to solve the problem:

1. **Vertical motion equation**: 
   Using the kinematic equation for vertical motion, the displacement \( y \) is:
   \[
   y = v_{0y} t + \frac{1}{2} g t^2
   \]
   where:
   - \( y = -h \) (because the skier moves downward by \( h \))
   - \( v_{0y} = v_0 \sin(\theta) \) (the vertical component of the velocity)
   - \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

   Thus, the equation becomes:
   \[
   -h = v_0 \sin(\theta) t - \frac{1}{2} g t^2
   \]

2. **Horizontal motion equation**:
   The horizontal displacement \( w \) can be found using:
   \[
   w = v_{0x} t
   \]
   where:
   - \( v_{0x} = v_0 \cos(\theta) \) (the horizontal component of the velocity)

3. **Solving for time \( t \)**:
   We first solve the vertical motion equation for time \( t \), then use that to calculate the horizontal distance \( w \).

Let's calculate this now.The maximum width \( w \) of the gorge that the skier can clear is approximately \( 41.78 \, \text{meters} \).

Would you like further details or clarification?

Here are 5 related questions to expand on this topic:
1. How would increasing the initial speed \( v_0 \) affect the width of the gorge the skier can clear?
2. What if the angle \( \theta \) were smaller or larger? How would that change the result?
3. How would air resistance affect the skier's motion and the gorge width?
4. Can you derive the maximum possible distance for a given height if the angle is optimized?
5. How can the energy conservation principle be applied to analyze the skier’s jump?

**Tip:** In projectile motion, the horizontal and vertical motions are independent of each other and can be analyzed separately using kinematic equations.

A secret agent skis off a slope inclined at θ = 31.5° with an initial speed of 16.6 m/s. The agent needs to clear a gorge, and the slope on the other side of the gorge is 17.1 meters below the starting point. What is the maximum width of the gorge that the agent clears?

This physics problem analyzes the projectile motion of a skier jumping off a slope. Learn how to calculate the maximum width of a gorge the skier can clear, using kinematic equations and trigonometry. Given the skier's speed of 16.6 m/s and a slope angle of 31.5°, find the solution step-by-step.

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