A body projected vertically from the earth reaches a height equal to earth's radius before returning to the earth. The power exerted by the gravitational force is greatest.

Options

A. at the instant just before the body hits the earth
B. it remains constant all through
C. at the instant just after the body is projected
D. at the highest position of the body

Answer

To solve the problem, we need to analyze the situation of a body projected vertically upward from the Earth, reaching a height equal to the Earth's radius, and then returning to the Earth. We are tasked with determining when the power exerted by the gravitational force is greatest.

### Step-by-Step Solution:

1. **Understanding Power in Relation to Gravitational Force**:
- The power \( P \) exerted by a force is given by the formula:
\[ P = \vec{F} \cdot \vec{V} = F V \cos \theta \]
where \( F \) is the magnitude of the gravitational force, \( V \) is the velocity of the body, and \( \theta \) is the angle between the force and the velocity vectors.

2. **Analyzing the Motion of the Body**:
- When the body is projected upwards, it initially has a positive velocity. As it rises, it decelerates due to the gravitational force acting downward.
- At the maximum height (which is equal to the Earth's radius), the velocity of the body becomes zero.

3. **Evaluating Power at Different Points**:
- **At the Highest Point**:
- The velocity \( V = 0 \), hence:
\[ P = F \cdot 0 = 0 \]
- Therefore, power is zero at the highest point.

- **Just After Projection**:
- The body is moving upwards with a positive velocity \( V \) while the gravitational force \( F \) acts downwards. The angle \( \theta = 180^\circ \):
\[ P = F V \cos(180^\circ) = -F V \]
- Power is negative, indicating that gravity is doing negative work (taking energy from the body).

- **Just Before Hitting the Earth**:
- The body is moving downwards with a positive velocity \( V \) and the gravitational force \( F \) is also acting downwards. The angle \( \theta = 0^\circ \):
\[ P = F V \cos(0^\circ) = F V \]
- Here, power is positive and maximum because both force and velocity are in the same direction.

4. **Conclusion**:
- The power exerted by the gravitational force is greatest just before the body hits the Earth, as both the gravitational force and the velocity are in the same direction at this point.

### Final Answer:
The power exerted by the gravitational force is greatest **just before the body hits the Earth** (Option 1).