https://file.aiquickdraw.com/mathbot/file/17268033537195f0dnk95.jpg

https://file.aiquickdraw.com/mathbot/file/1726803353723wdyqich7.jpg

To solve the given problem involving the ski jumper:

### Problem Breakdown:

1. **Horizontal motion**: The ski jumper moves horizontally off the slope with a speed of 25 m/s.
2. **Slope angle**: The landing slope has an angle \(\theta = 52^\circ\).
3. **Acceleration due to gravity**: \(g = 9.8 \, \text{m/s}^2\).

We need to determine:

1. **(Part 2)**: How long the ski jumper is airborne.
2. **(Part 3)**: The relative angle \(\phi\) at which the ski jumper hits the slope.

---

### Part 2: Time of Flight

The motion can be separated into horizontal and vertical components. Since the skier leaves the track horizontally, the initial vertical velocity is 0.

- **Horizontal motion**: The horizontal velocity \(v_x = 25 \, \text{m/s}\) remains constant because there's no air resistance.
  
- **Vertical motion**: The only force acting on the skier vertically is gravity, so we use the following kinematic equation for the vertical displacement:
  \[
  y = \frac{1}{2} g t^2
  \]
  where \(y\) is the vertical distance fallen and \(t\) is the time in seconds.

The slope of the landing surface is given by \(\theta = 52^\circ\). To find the time of flight, we note that the jumper hits the slope when the coordinates \((x, y)\) satisfy the equation of the slope:
\[
y = x \tan \theta
\]
The horizontal distance \(x\) covered by the jumper in time \(t\) is given by:
\[
x = v_x t = 25t
\]

Substituting into the slope equation:
\[
y = 25t \tan(52^\circ)
\]

From the vertical motion equation:
\[
y = \frac{1}{2} g t^2 = \frac{1}{2} (9.8) t^2
\]

Equating the two expressions for \(y\):
\[
25t \tan(52^\circ) = \frac{1}{2} (9.8) t^2
\]

Solve this equation for \(t\) to find how long the skier is airborne. I'll calculate the value now.The time the ski jumper is airborne is approximately **6.53 seconds**.

---

### Part 3: Relative Angle of Impact

To find the angle \(\phi\) with which the jumper hits the slope, we need to analyze the velocity components:

1. **Horizontal velocity**: \(v_x = 25 \, \text{m/s}\) (remains constant).
2. **Vertical velocity**: This increases due to gravity and can be found using:
   \[
   v_y = g t
   \]
   where \(t = 6.53 \, \text{s}\) (the time in the air).

The total velocity \(v\) upon impact forms an angle \(\alpha\) with the horizontal, which is given by:
\[
\alpha = \tan^{-1} \left(\frac{v_y}{v_x}\right)
\]

Finally, the relative angle \(\phi\) at which the jumper hits the slope is the difference between the impact angle \(\alpha\) and the slope angle \(\theta\):
\[
\phi = \alpha - \theta
\]

I'll calculate the impact angle \(\alpha\) and the relative angle \(\phi\) next.The relative angle \(\phi\) at which the ski jumper hits the slope is approximately **16.66°**.

---

Would you like any further details or clarifications on this solution?

Here are five related questions for further exploration:

1. How would air resistance affect the time the jumper is airborne?
2. What is the impact speed (magnitude of the velocity) of the jumper upon landing?
3. How would a steeper slope (e.g., \( \theta = 60^\circ \)) affect the time of flight?
4. How does the horizontal speed affect the angle \(\phi\) of impact?
5. What would happen if the jumper left the slope with a vertical component in their velocity?

**Tip**: When solving projectile motion problems, separating the horizontal and vertical components of motion is a key step for simplicity.

A ski jumper travels down a slope and leaves the ski track moving in the horizontal direction with a speed of 25 m/s. The landing incline has a slope of θ = 52°. The acceleration of gravity is 9.8 m/s². Determine how long the ski jumper is airborne and the relative angle φ with which the ski jumper hits the slope.

Solve the ski jumper problem involving horizontal motion, slope angle of 52°, and gravity. Learn how to calculate the airborne time and the relative impact angle φ using kinematic equations and trigonometry. Suitable for college-level or advanced high school physics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation