A ball is projected from the bottom of an inclined plane of inclination `30^@`, with a velocity of `30 ms^(-1)`, at an angle of `30^@` with the inclined plane. If `g = 10 ms^(-2)` , then the range of the ball on given inclined plane is

Options

A. 12 m
B. 60 m
C. 120 m
D. 600 m

Answer

To solve the problem of finding the range of a ball projected from the bottom of an inclined plane, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Angle of inclination of the plane, \( \theta = 30^\circ \)
- Initial velocity of the ball, \( v = 30 \, \text{m/s} \)
- Acceleration due to gravity, \( g = 10 \, \text{m/s}^2 \)
- Angle of projection with respect to the inclined plane, \( \alpha = 30^\circ \)

2. **Resolve the Initial Velocity**:
- The component of the initial velocity along the inclined plane is given by:
\[ v_{\text{along}} = v \cos \alpha = 30 \cos 30^\circ = 30 \times \frac{\sqrt{3}}{2} = 15\sqrt{3} \, \text{m/s} \]

3. **Determine the Vertical Component of Gravity**:
- The effective acceleration due to gravity acting down the incline is:
\[ g_{\text{effective}} = g \sin \theta = 10 \sin 30^\circ = 10 \times \frac{1}{2} = 5 \, \text{m/s}^2 \]

4. **Use the Range Formula for Projectile Motion on an Incline**:
- The range \( R \) on an inclined plane can be calculated using the formula:
\[ R = \frac{v_{\text{along}}^2}{g_{\text{effective}}} \sin(2\alpha) \]
- Here, \( \sin(2\alpha) = \sin(60^\circ) = \frac{\sqrt{3}}{2} \).

5. **Substituting the Values**:
- Substitute \( v_{\text{along}} \) and \( g_{\text{effective}} \) into the range formula:
\[ R = \frac{(15\sqrt{3})^2}{5} \cdot \frac{\sqrt{3}}{2} \]
- Calculate \( (15\sqrt{3})^2 = 675 \):
\[ R = \frac{675}{5} \cdot \frac{\sqrt{3}}{2} = 135 \cdot \frac{\sqrt{3}}{2} = 67.5\sqrt{3} \, \text{m} \]

6. **Final Calculation**:
- Approximating \( \sqrt{3} \approx 1.732 \):
\[ R \approx 67.5 \times 1.732 \approx 116.5 \, \text{m} \]

### Conclusion:
The range of the ball on the inclined plane is approximately \( 116.5 \, \text{m} \).