What is kinetic energy ? Derive an expressions for the kinetic energy of a body of mass 'm' moving at a speed 'v'.

Explanation

Kinetic energy :
The energy possessed by an object due to its motion is called kinetic energy.
Numerical expression for K.E. :
1) Let us assume that an object of mass (m) is at rest on a smooth horizontal plane as shown in figure.

2) Let it be displaced through a distance 's' from the point A to B by a force (F) acting upon it in the direction of the displacement.
3) In the horizontal direction the net force `'F_("net")'` is equal to the force applied 'F'.
`therefore W=F_("net").s=F.s " "` ............ (1)
4) Let the work done on the object cause a change in its velocity form 'u' to 'v' and the 'acceleration produced be 'a'.
5) We know that `v^(2)-u^(2)=2as rArrs=(v^(2)-u^(2))/(2a) " " ............(2)`
6) From Newton's second law of motion, F = ma .............. (3)
7) From (1), (2) and (3)
`W=(ma(v^(2)-u^(2)))/(2a)`
`W=1/2m(v^(2)-u^(2))`
8) As we have assumed that object is at rest, then the initial velocity u= 0
`therefore W=1/2mv^(2)`
9) We know that K.E. of a body moving with certain velocity is equal to work done on the object to acquire that velocity from rest.
10) Thus the K.E. of an object of mass 'm' and moving with velocity 'v' is equal to
`K.E.=1/2mv^(2)`