A ball falls freely form a height onto and smooth inclined plane forming an angle a with the horizontal. Find the ratio of the distance between the points at which the jumping ball strikes the inclined plane. Assume the impacts to be elastic.

Explanation

To solve the problem of a ball falling freely from a height onto a smooth inclined plane, we need to analyze the motion of the ball and the impacts it makes with the inclined plane. Let's break down the solution step by step.

### Step 1: Understanding the Motion of the Ball
The ball falls freely under the influence of gravity from a height \( h \) onto the inclined plane, which makes an angle \( \alpha \) with the horizontal. The initial velocity of the ball just before it strikes the inclined plane can be determined using the equations of motion.

**Hint:** Remember that the initial velocity of the ball just before impact can be found using the kinematic equations.

### Step 2: Resolving the Velocity Components
At the moment of impact, the velocity \( v_0 \) of the ball can be resolved into two components:
- Perpendicular to the inclined plane: \( v_{0y} = v_0 \cos \alpha \)
- Parallel to the inclined plane: \( v_{0x} = v_0 \sin \alpha \)

**Hint:** Use trigonometric functions to resolve the velocity into components based on the angle of inclination.

### Step 3: Analyzing the Impact
Since the impact is elastic, the component of velocity perpendicular to the inclined plane will reverse its direction, while the parallel component will remain unchanged. Therefore, after the impact:
- The new perpendicular velocity: \( v_{1y} = -v_{0y} = -v_0 \cos \alpha \)
- The new parallel velocity: \( v_{1x} = v_{0x} = v_0 \sin \alpha \)

**Hint:** Recall that in an elastic collision, the speed is conserved, but the direction of the perpendicular component changes.

### Step 4: Time of Flight Calculation
To find the distance traveled by the ball after each impact, we need to calculate the time of flight until it strikes the inclined plane again. The time of flight can be calculated using the kinematic equations.

Using the equations of motion, we can find the time \( t \) for the ball to travel a distance \( d \) along the inclined plane after the first impact.

**Hint:** Use the equation \( d = v_{1x} t + \frac{1}{2} a_x t^2 \) to find the time of flight.

### Step 5: Finding Distances After Each Impact
The distances \( r_1, r_2, r_3 \) that the ball travels after each impact can be calculated using the respective time of flight and the velocity components.

1. For the first impact, the distance \( r_1 \) can be calculated as:
\[ r_1 = v_{0x} t_1 + \frac{1}{2} a_x t_1^2 \]
2. For the second impact, the distance \( r_2 \) can be calculated similarly, but using the new velocities after the first impact.
3. For the third impact, repeat the process.

**Hint:** Remember to substitute the values of \( v_{1x} \) and \( a_x \) correctly for each subsequent impact.

### Step 6: Ratio of Distances
Finally, we need to find the ratio of the distances \( r_1 : r_2 : r_3 \). After calculating the distances, we can express them in terms of a common factor and simplify to find the ratio.

**Hint:** Factor out common terms to simplify the ratio of the distances.

### Conclusion
After performing the calculations, we find that the ratio of the distances at which the ball strikes the inclined plane is:
\[ r_1 : r_2 : r_3 = 1 : 2 : 3 \]

This concludes the solution to the problem.