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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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