Two bodies slide from rest down two smooth inclined planes commencing at the same point and terminating in the same horizontal plane. The ratio of the velocities attained if inclinations to the horizontal of the planes are `30^(@)` and `60^(@)` respectively is

Options

A. `1:sqrt(3)`
B. `2: sqrt(3)`
C. `1:1`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the velocities attained by two bodies sliding down two smooth inclined planes with angles of inclination \(30^\circ\) and \(60^\circ\).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two bodies sliding down two inclined planes.
- The angles of inclination are \(30^\circ\) and \(60^\circ\).
- Both bodies start from rest and slide down to the same horizontal level.

2. **Using the Formula for Final Velocity on an Inclined Plane**:
- The final velocity \(v\) of an object sliding down an incline can be derived from energy conservation or kinematics. The formula is:
\[ v = \sqrt{2gh \sin(\theta)} \]
- Here, \(h\) is the height from which the body descends, \(g\) is the acceleration due to gravity, and \(\theta\) is the angle of inclination.

3. **Calculating the Final Velocities**:
- For the first body on the \(30^\circ\) incline:
\[ v_1 = \sqrt{2gh \sin(30^\circ)} = \sqrt{2gh \cdot \frac{1}{2}} = \sqrt{gh} \]
- For the second body on the \(60^\circ\) incline:
\[ v_2 = \sqrt{2gh \sin(60^\circ)} = \sqrt{2gh \cdot \frac{\sqrt{3}}{2}} = \sqrt{gh\sqrt{3}} = \sqrt{gh} \cdot \sqrt{\sqrt{3}} = \sqrt{gh} \cdot \frac{\sqrt{3}}{2} \]

4. **Finding the Ratio of Velocities**:
- Now, we can find the ratio of the velocities \(v_1\) and \(v_2\):
\[ \frac{v_1}{v_2} = \frac{\sqrt{gh}}{\sqrt{gh\sqrt{3}}} = \frac{\sqrt{gh}}{\sqrt{gh} \cdot \sqrt{\sqrt{3}}} = \frac{1}{\sqrt{\sqrt{3}}} = \frac{1}{\sqrt{3^{1/2}}} = \frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]

5. **Final Result**:
- The ratio of the velocities attained by the two bodies is:
\[ \frac{v_1}{v_2} = \frac{1}{\sqrt{3}} \text{ or } \frac{\sqrt{3}}{3} \]