Question

Derive the formula for the kinetic energy of an object of mass m, moving with velocity v.

Answer 1

Let the initial velocity of the object be u. Let an external force be applied to it so that it gets displaced by distance s and its velocity becomes v. In this scenario, the kinetic energy of the moving body is equal to the work that was required to change its velocity from u to v.

Thus, we have the velocity-position relation as:

v² = u² + 2as

or

`s = (ν^2 - u^2)/(2a)`   ....(i)

Where a is the acceleration of the body during the change in its velocity

Now, the work done on the body by the external force is given by:

W = F × s

F = ma …(ii)

From equations (i) and (ii), we obtain,
`W = ma xx ((ν^2 - u^2)/(2a)) = 1/2 m(ν^2 - u^2)`

If the body was initially at rest (i.e., u =0), then
`W = 1/2mν^2 `
Since kinetic energy is equal to the work done on the body to change its velocity from 0 to v, we obtain:
Kinetic energy, `E_k = 1/2mν^2`