Hailstrom are observed to strike the surface of a frozen lake in a direction making an angle of `30^(@)` to the vertical and to rebound at an angle of `60^(@)` to the vertical. Assuming the contact to be smooth, the coefficient of restitution is

Options

A. `1//3`
B. `2//3`
C. `1//sqrt(3)`
D. None of these

Answer

To find the coefficient of restitution (e) for the hailstones striking and rebounding off the surface of a frozen lake, we can follow these steps:

### Step 1: Understand the Angles
The hailstones strike the surface at an angle of 30° to the vertical. This means the angle to the horizontal (the surface of the lake) is:
\[ \theta_1 = 90° - 30° = 60° \]
Similarly, they rebound at an angle of 60° to the vertical, which means the angle to the horizontal is:
\[ \theta_2 = 90° - 60° = 30° \]

### Step 2: Resolve the Velocities
Let’s denote the velocities of the hailstones just before impact and just after rebounding as \(v_1\) and \(v_2\) respectively. We need to resolve these velocities into components parallel and perpendicular to the surface of the lake.

The vertical component of the velocity before impact is:
\[ v_{1y} = v_1 \cos(60°) = v_1 \cdot \frac{1}{2} \]
The horizontal component of the velocity before impact is:
\[ v_{1x} = v_1 \sin(60°) = v_1 \cdot \frac{\sqrt{3}}{2} \]

The vertical component of the velocity after rebounding is:
\[ v_{2y} = v_2 \cos(30°) = v_2 \cdot \frac{\sqrt{3}}{2} \]
The horizontal component of the velocity after rebounding is:
\[ v_{2x} = v_2 \sin(30°) = v_2 \cdot \frac{1}{2} \]

### Step 3: Apply the Coefficient of Restitution Formula
The coefficient of restitution (e) is defined as the ratio of the relative speed after the collision to the relative speed before the collision, specifically in the direction perpendicular to the surface:
\[ e = \frac{\text{Relative speed after}}{\text{Relative speed before}} = \frac{v_{2y}}{v_{1y}} \]

### Step 4: Substitute the Components
Substituting the vertical components into the equation:
\[ e = \frac{v_2 \cdot \frac{\sqrt{3}}{2}}{v_1 \cdot \frac{1}{2}} = \frac{v_2 \cdot \sqrt{3}}{v_1} \]

### Step 5: Find the Relationship Between \(v_1\) and \(v_2\)
Since the hailstones rebound at an angle, we can relate \(v_1\) and \(v_2\) using the angles:
From the conservation of momentum in the vertical direction (assuming no energy loss in the horizontal direction due to smooth contact):
\[ v_{1y} = e \cdot v_{2y} \]
Substituting the expressions for \(v_{1y}\) and \(v_{2y}\):
\[ v_1 \cdot \frac{1}{2} = e \cdot v_2 \cdot \frac{\sqrt{3}}{2} \]
Rearranging gives:
\[ e = \frac{v_1}{v_2} \cdot \frac{1}{\sqrt{3}} \]

### Step 6: Calculate the Coefficient of Restitution
From the angles, we can see that:
\[ \frac{v_1}{v_2} = \frac{\sin(60°)}{\sin(30°)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]
Now substituting back into the equation for \(e\):
\[ e = \sqrt{3} \cdot \frac{1}{\sqrt{3}} = 1 \]

### Final Answer
The coefficient of restitution is:
\[ \boxed{1} \]