### WORK-ENERGY RELATION

WORK ENERGY EQUATION

DERIVATION:
Let us consider a body of mass “m” is placed at point A at a height h from the surface of earth.At this point the body possesses gravitational potential energy equal to mgh w.r.t point C lying on the ground.
Now consider a point B at a distance x below the point A during downward motion of body.At this stage the height of the body becomes (h-x).

so, potential energy at point B becomes,
P.E=mg(h-x)
As we know that potential energy at point B is less than the potential energy at point A,i.e.
mg(h-x) < mgh
or                                 mgh – mgx <mgh

The loss in potential energy at point B is mgx.

The Kinetic Energy at point A is equal to zero because the body is at rest.During its downward motion its velocity increases ,so its kinetic energy also increases.If there is no air friction then the loss of P.E is equal to the gain in K.E, means P.E is converted into K.E.

When the body reaches at point C its P.E becomes zero which means all of its P.E is converted into K.E,so we can write as
Loss in P.E = Gain in K.E

Practically there is always a force of friction which opposes the downward motion of the body.Let if friction f is present in this case then some amount of P.E is lost in work done against friction.Now, the modified equation can be written as,

Loss in P.E = Gain in K.E + Work done against friction
mgx= 1/2 mv2 + fx
or                                 1/2 mv2 = mgx – fx

In terms of “h”
1/2 mv2 = mgh – fh

The above equation is known as “Work Energy Equation”.