Problem 3:   A secret agent skis off a slope inclined at θ = 25.9 degrees below horizontal at a speed of v0 = 13.8 m/s. He must clear a gorge, and the slope on the other side of the gorge is h = 13 m below the edge of the upper slope.

What is the maximum width, w, of the gorge (in meters) so that the agent clears it?
Numeric   : A numeric value is expected and not an expression.
w = __________________________________________

This problem involves projectile motion as the secret agent skis off the slope. We'll use the principles of projectile motion to find the maximum width \( w \) of the gorge.

### Given:
- Initial speed \( v_0 = 13.8 \, \text{m/s} \)
- Angle of inclination \( \theta = 25.9^\circ \)
- Vertical drop \( h = 13 \, \text{m} \)
- Gravitational acceleration \( g = 9.8 \, \text{m/s}^2 \)

We can break down the problem as follows:

1. **Vertical motion:**
   The vertical distance traveled by the agent is \( h = 13 \, \text{m} \). The vertical component of the initial velocity is:
   \[
   v_{0y} = v_0 \sin(\theta)
   \]
   Using the kinematic equation for vertical displacement:
   \[
   h = v_{0y} t + \frac{1}{2} g t^2
   \]
   Since the agent is moving downward, the displacement \( h \) is positive, and we use a negative value for \( v_{0y} \), since it points downward. Solving this equation will give us the time \( t \) it takes for the agent to clear the gorge.

2. **Horizontal motion:**
   The horizontal distance traveled (the width of the gorge) is:
   \[
   w = v_{0x} t
   \]
   where \( v_{0x} = v_0 \cos(\theta) \).

Let’s compute the required values.The maximum width \( w \) of the gorge that the secret agent can clear is approximately \( 13.98 \, \text{m} \).

Would you like more details or have any questions?

Here are five related questions to expand your understanding:
1. How would the width of the gorge change if the initial speed was increased?
2. How does the angle of the slope impact the distance the agent can travel?
3. What happens to the width if the vertical drop \( h \) is reduced to 10 m?
4. How would air resistance affect the agent's ability to clear the gorge?
5. Can this problem be solved using energy conservation principles?

**Tip:** Always break the motion into horizontal and vertical components when solving projectile motion problems.

Problem 3: A secret agent skis off a slope inclined at θ = 25.9 degrees below horizontal at a speed of v0 = 13.8 m/s. He must clear a gorge, and the slope on the other side of the gorge is h = 13 m below the edge of the upper slope. What is the maximum width, w, of the gorge (in meters) so that the agent clears it? w = __________________________________________

This problem involves using principles of projectile motion to calculate the maximum width a secret agent can clear while skiing off a slope inclined at 25.9 degrees. Given an initial speed of 13.8 m/s and a vertical drop of 13 meters, we determine the maximum width of the gorge using kinematic equations. Ideal for high school students in Grades 10-12.

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