A body projected vertically from the earth reaches a height equal to earth’s radius before returning to the ea

Question

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:

Accepted Answer

1. **Definition of Power:** Power is the scalar product of force and velocity, given by $P = \mathbf{F} \cdot \mathbf{v} = Fv \cos 	heta$, where $	heta$ is the angle between the force and velocity vectors.
2. **Gravitational Force:** The force acting on the body is $F = \frac{GMm}{r^2}$, directed towards the center of the Earth. Its magnitude is maximum at the surface ($r=R$) and decreases with height.
3. **Velocity Analysis:** 
   - At the instant of projection (surface), velocity $v$ is maximum. Angle $	heta = 180^\circ$ (force down, velocity up). Power $P = -F_{max}v_{max}$.
   - At maximum height, velocity $v = 0$. Power $P = 0$.
   - Just before hitting the Earth (surface), velocity $v$ is maximum. Angle $	heta = 0^\circ$ (force down, velocity down). Power $P = +F_{max}v_{max}$.
4. **Comparison:** The power is negative during ascent and positive during descent. Algebraically, the maximum value occurs when the power is positive and the product of force and velocity is maximized. This happens at the surface just before impact.

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