A body of mass 2 kg is projected at 20 `ms^(-1)` at an angle `60^(@)`above the horizontal. Power due to the gravitational force at its highest point is

Options

A. 200 W
B. `100sqrt3` W
C. 50 W
D. zero

Explanation

To solve the problem, we need to determine the power due to the gravitational force acting on a body at its highest point during its projectile motion.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have a body of mass \( m = 2 \, \text{kg} \) projected at a speed of \( u = 20 \, \text{m/s} \) at an angle of \( \theta = 60^\circ \) above the horizontal.

2. **Finding the Components of Initial Velocity:**
- The initial velocity can be broken down into horizontal and vertical components:
- \( u_x = u \cos \theta = 20 \cos 60^\circ = 20 \times \frac{1}{2} = 10 \, \text{m/s} \)
- \( u_y = u \sin \theta = 20 \sin 60^\circ = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3} \, \text{m/s} \)

3. **Determining the Highest Point:**
- At the highest point of its trajectory, the vertical component of the velocity (\( v_y \)) becomes 0. The horizontal component (\( v_x \)) remains constant throughout the motion:
- \( v_x = u_x = 10 \, \text{m/s} \)
- \( v_y = 0 \, \text{m/s} \)

4. **Calculating the Gravitational Force:**
- The gravitational force acting on the body is given by:
- \( F_g = m \cdot g = 2 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 19.62 \, \text{N} \)
- This force acts downwards.

5. **Calculating Power:**
- Power (\( P \)) due to a force is given by the formula:
- \( P = F \cdot v \cdot \cos(\phi) \)
- Here, \( F \) is the gravitational force, \( v \) is the velocity of the body at the highest point, and \( \phi \) is the angle between the force and the velocity vector.
- At the highest point, the velocity vector is horizontal, and the gravitational force is vertical. Therefore, the angle \( \phi = 90^\circ \).
- Since \( \cos(90^\circ) = 0 \):
- \( P = F_g \cdot v \cdot \cos(90^\circ) = 19.62 \cdot 10 \cdot 0 = 0 \, \text{W} \)

### Final Answer:
The power due to the gravitational force at the highest point is \( 0 \, \text{W} \).