Question

A body is projected vertically upwards from earth's surface. If its K.E. of projection is equal to half of its minimum value required to escape from the gravitational influence, then the height upto which it rises is (R = radius of the earth)

• A. 4R
• B. R
• C. 2R
• D. 3R

Hint:

The height reached by a body projected upwards depends on its initial kinetic energy. If the kinetic energy is half of the escape energy, the body will rise to the Earth's radius.

Answer 1

The Correct Option is B

Step 1: Understanding the energy required to escape.
The minimum kinetic energy required for a body to escape from the Earth's gravitational field is equal to the gravitational potential energy at the Earth's surface. The total energy at the surface is the sum of the kinetic energy and potential energy: \[ E = \frac{1}{2} m v^2 - \frac{GMm}{R} \] For escape, this total energy must be zero. If the kinetic energy is half of this value, then the body will rise to a height equal to the radius of the Earth, \( R \), before it comes to rest.
Step 2: Conclusion.
Thus, the height upto which the body rises is \( R \), which is option (B).