A particle is fired horizontally form an inclined plane of inclination 30 ∘ with horizontal with speed 50 m s − 1 . If g = 10 m s − 2 , the range measured along the incline is

Answer

To find the range of a particle fired horizontally from an inclined plane, we can use the formula for range along an incline:

\[ R = \frac{u^2}{g \cos^2 \beta} \left( \sin(2\alpha + \beta) + \sin(\beta) \right) \]

Where:
- \( R \) is the range along the incline,
- \( u \) is the initial speed,
- \( g \) is the acceleration due to gravity,
- \( \alpha \) is the angle of projection (which is 0 degrees in this case since the particle is fired horizontally),
- \( \beta \) is the angle of inclination of the plane.

### Step 1: Identify the given values
- \( u = 50 \, \text{m/s} \)
- \( g = 10 \, \text{m/s}^2 \)
- \( \beta = 30^\circ \)
- \( \alpha = 0^\circ \)

### Step 2: Calculate \( \cos^2 \beta \)
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \implies \cos^2(30^\circ) = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 3: Calculate \( \sin(2\alpha + \beta) \)
Since \( \alpha = 0^\circ \):
\[ 2\alpha + \beta = 2(0) + 30^\circ = 30^\circ \]
\[ \sin(30^\circ) = \frac{1}{2} \]

### Step 4: Calculate \( \sin(\beta) \)
\[ \sin(30^\circ) = \frac{1}{2} \]

### Step 5: Substitute values into the range formula
Now substituting all the values into the range formula:
\[ R = \frac{u^2}{g \cos^2 \beta} \left( \sin(2\alpha + \beta) + \sin(\beta) \right) \]
\[ R = \frac{50^2}{10 \cdot \frac{3}{4}} \left( \frac{1}{2} + \frac{1}{2} \right) \]
\[ R = \frac{2500}{10 \cdot \frac{3}{4}} \cdot 1 \]
\[ R = \frac{2500}{\frac{30}{4}} = \frac{2500 \cdot 4}{30} = \frac{10000}{30} = \frac{1000}{3} \, \text{m} \]

### Final Answer
The range measured along the incline is:
\[ R = \frac{1000}{3} \, \text{m} \approx 333.33 \, \text{m} \]