A particle moves in a straight line with retardatino proportional it its displacement. Calculate the loss of `K.E.` for any displacement `x`.

Explanation

Here, retardation `prop` displacement
`-a prop x ` or `-a=Kx`
where `K` is constant of proportionality.
From (i) `as=-(dupsilon)/(dt)=Kx` or `(dupsilon)/(dt)(dx)/(dx)=-Kx`
As `(dx)/(dt)=upsilon`, therefore, `upsilon dx=-Kx dx`
`int_(0)^(upsilon)upsilon dx=-Kint_(0)^(x)x dx`
or `[(upsilon^(2))/(2)]_(u)^(upsilon)=-K[(x^(2))/(2)]_(0)^(x)`
or `(1)/(2)m upsilon^(2)-(1)/(2)m u^(2)=-(Kx^(2)xxm)/(2)`
Thus loss in `K.E., DeltaK=-(Kmx^(2))/(2)`