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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):


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Solution

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, PE=12mva2-12mvb2, where m is the mass of the body, va and vb 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, PE=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, PE=12mva2-12mvb2.

Now,

mgR=12mv2-0(since,atearthsurface,v=0)orv2=2gRorv=2gR

Therefore, the velocity of the body on reaching the earth is v=2gR.


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