The maximum range of rifle bullet on the horizontal ground is 6 km its maximum range on an inclined of 30° will be

Correct Option: 4 km

To find the maximum range of the projectile on an inclined plane, we can relate it to the maximum range on a horizontal surface.

Step 1: Understand the maximum range on horizontal ground
The formula for the range of a projectile on horizontal ground is given by:

$R=\frac{u^2\sin \left(2\theta \right)}{g}$

where $u$ is the initial velocity of projection, $\theta$ is the angle of projection, and $g$ is the acceleration due to gravity.
The range is maximum when $\sin \left(2\theta \right)=1$ (i.e., $\theta ={45}^{\circ }$):

$R_{\text{max}}=\frac{u^2}{g}=6\text{km}$

Step 2: Use the formula for maximum range up an inclined plane
The maximum range of a projectile projected up an inclined plane of inclination angle $\beta$ is given by:

$R_{\text{inclined}}=\frac{u^2}{g(1+\sin \beta )}$

Here, the inclination of the plane is given as $\beta ={30}^{\circ }$.

Step 3: Calculate the maximum range on the inclined plane
Substitute the values into the formula:

$R_{\text{inclined}}=\frac{R_{\text{max}}}{1+\sin \left({30}^{\circ }\right)}$

Since $\sin \left({30}^{\circ }\right)=0.5$ and $R_{\text{max}}=6\text{km}$:

$R_{\text{inclined}}=\frac{6}{1+0.5}=\frac{6}{1.5}=4\text{km}$

Therefore, the maximum range of the rifle bullet on the 30° inclined plane is 4 km.