The slopes of wind screen of two cars are `alpha_1=30^@ and alpha_2= 15^@` respectively. At what ratio `v_1/ v_2` of the velocities of the cars will their drivers see the hail stones bounced back by the wind screen on their cars in vertical direction? Assume hail stones fall vertically downwards and collisions to be elastic.

Explanation

To solve the problem, we need to determine the ratio of the velocities \( v_1/v_2 \) of two cars such that the drivers see hailstones bouncing back from their windscreens in a vertical direction. The slopes of the windscreens are given as \( \alpha_1 = 30^\circ \) and \( \alpha_2 = 15^\circ \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Hailstones fall vertically downwards with a velocity \( v \).
- The cars are moving with velocities \( v_1 \) and \( v_2 \).
- The angle of incidence \( \beta_1 \) for the hailstone on the first car's windscreen and \( \beta_2 \) for the second car's windscreen can be related to the slopes \( \alpha_1 \) and \( \alpha_2 \).

2. **Setting Up the Geometry**:
- For the first car, the slope of the windscreen is \( \alpha_1 \), and for the second car, it is \( \alpha_2 \).
- The angle of incidence \( \beta_1 \) can be expressed as:
\[ \beta_1 = 90^\circ - \alpha_1 = 90^\circ - 30^\circ = 60^\circ \]
- Similarly, for the second car:
\[ \beta_2 = 90^\circ - \alpha_2 = 90^\circ - 15^\circ = 75^\circ \]

3. **Using the Law of Reflection**:
- For the hailstones to bounce back vertically, the angle of incidence must equal the angle of reflection. Therefore, we have:
\[ \alpha_1 + 2\beta_1 = 90^\circ \quad \text{and} \quad \alpha_2 + 2\beta_2 = 90^\circ \]

4. **Relating Velocities**:
- The relationship between the velocities and angles can be derived from the tangent function:
\[ \tan(\alpha) = \frac{v}{v_i} \]
- For the first car:
\[ \tan(\alpha_1) = \frac{v}{v_1} \]
- For the second car:
\[ \tan(\alpha_2) = \frac{v}{v_2} \]

5. **Finding the Ratio of Velocities**:
- We can express the ratio of the velocities as:
\[ \frac{v_1}{v_2} = \frac{\tan(\alpha_2)}{\tan(\alpha_1)} \]
- Substituting the angles:
\[ \frac{v_1}{v_2} = \frac{\tan(15^\circ)}{\tan(30^\circ)} \]

6. **Calculating the Tangents**:
- We know:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \quad \text{and} \quad \tan(15^\circ) = 2 - \sqrt{3} \]
- Thus:
\[ \frac{v_1}{v_2} = \frac{2 - \sqrt{3}}{\frac{1}{\sqrt{3}}} = (2 - \sqrt{3}) \cdot \sqrt{3} = 2\sqrt{3} - 3 \]

7. **Final Result**:
- The ratio of the velocities \( v_1/v_2 \) is approximately \( 3:1 \).