### 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 mv

^{2}+ fx
or 1/2 mv

^{2}= mgx – fx
In terms
of “h”

1/2 mv

^{2}= mgh – fh
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