Question

A small body of mass $m$ falls to the earth from an infinite distance away. What will be its velocity on reaching the earth? (Radius of the earth is $R$, acceleration due to gravity on the surface of the earth is $g$):

Answer 1

Step 1: Given data

The radius of the earth is $R$.

The acceleration due to gravity of the earth is $g$.

Step 2: Potential energy and formula used

1. The potential energy of a body at point P is defined as the total work done to bring a body from infinity to that point. In simple words, potential energy is the change in kinetic energy between two places.
2. The potential energy at a point is defined by the form, $P_E=\left(\frac{1}{2}m{v_a}^2-\frac{1}{2}m{v_b}^2\right)$, where m is the mass of the body, $v_a$ and $v_b$ are the velocity of the points at a and b.
3. Again we know that potential energy at a height R is defined by the form, $P_E=mgR$.

Step 3: Finding the velocity

Let v be the velocity of the body before reaching the earth.

As we know, potential energy is the change in kinetic energy. So, $P_E=\left(\frac{1}{2}m{v_a}^2-\frac{1}{2}m{v_b}^2\right)$.

Now,

$$\begin{gathered} mgR=\frac{1}{2}m{v}^2-0(\sin ce,atearthsurface,v=0) \\ or{v}^2=2gR \\ orv=\sqrt{2gR} \end{gathered}$$

Therefore, the velocity of the body on reaching the earth is $v=\sqrt{2gR}$.