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A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to (i) t1/2 (ii) t (iii) t3/2 (iv) t2

Q10  A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to 

         (i) t^{1/2}\: \: (ii) t \: \: (iii) t^{3/2}\: \: (iv) t^2

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We know that the power is given by : 

                                                P\ =\ F.v

or                                                    =\ ma.v

or                                                   =\ m\frac{dv}{dt}v

It is given that power is constant, thus :

                                             mv\frac{dv}{dt}\ =\ constant

or                                              vdv\ =\ \frac{C}{m}dt

By integrating both sides, we get

                                                  v\ =\ \left ( \sqrt{\frac{2Ct}{m}} \right )

Also, we can write :

                                                 v\ =\ \frac{dx}{dt}

or                                              \frac{dx}{dt}\ =\ \sqrt{\frac{2C}{m}}t^\frac{1}{2}

By integrating we get the relation :

                                                 x\ \propto \ \ t^\frac{3}{2}

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