A stone is projected at angle `30^(@)` to the horizontal.The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of fight will be -

Options

A. `4:1`
B. `1:2`
C. `4:3`
D. `1:4`

Answer

To solve the problem, we need to find the ratio of the kinetic energy of a stone at the point of projection to its kinetic energy at the highest point of its flight when projected at an angle of \(30^\circ\) to the horizontal.

### Step-by-Step Solution:

1. **Identify the Initial Kinetic Energy (KE1):**
The initial kinetic energy of the stone when it is projected can be expressed as:
\[ KE_1 = \frac{1}{2} m v^2 \]
where \(m\) is the mass of the stone and \(v\) is the initial velocity of the stone.

2. **Determine the Velocity at the Highest Point (v_y):**
At the highest point of the projectile's flight, the vertical component of the velocity becomes zero. The horizontal component of the velocity remains unchanged. The initial velocity can be broken down into its horizontal and vertical components:
\[ v_x = v \cos(30^\circ) = v \cdot \frac{\sqrt{3}}{2} \]
\[ v_y = v \sin(30^\circ) = v \cdot \frac{1}{2} \]
At the highest point, the vertical component \(v_y\) is \(0\), and only the horizontal component \(v_x\) contributes to the kinetic energy.

3. **Calculate the Kinetic Energy at the Highest Point (KE2):**
The kinetic energy at the highest point can be expressed as:
\[ KE_2 = \frac{1}{2} m v_x^2 = \frac{1}{2} m \left(v \cdot \frac{\sqrt{3}}{2}\right)^2 \]
Simplifying this gives:
\[ KE_2 = \frac{1}{2} m \cdot \frac{3}{4} v^2 = \frac{3}{8} m v^2 \]

4. **Find the Ratio of Kinetic Energies (KE1/KE2):**
Now, we can find the ratio of the initial kinetic energy to the kinetic energy at the highest point:
\[ \text{Ratio} = \frac{KE_1}{KE_2} = \frac{\frac{1}{2} m v^2}{\frac{3}{8} m v^2} \]
The mass \(m\) and \(v^2\) cancel out:
\[ \text{Ratio} = \frac{\frac{1}{2}}{\frac{3}{8}} = \frac{1}{2} \cdot \frac{8}{3} = \frac{4}{3} \]

### Final Answer:
The ratio of the kinetic energy of the stone at the point of projection to its kinetic energy at the highest point of flight is:
\[ \frac{4}{3} \]