Ncert Exercises & Solutions

Question 6.10:

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

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

(iii) $t^{1 / 2}$

Power is given by the relation:

P = Fv

= mav = mv $\frac{dv}{dt} = Constant (say, k)$

$\Rightarrow vdv = \frac{k}{m} dt$

Integrating both sides:

$\int v d v = \int \frac{k}{m} d t$
$\frac{v^{2}}{2} = \frac{k}{m} t$
$v = \sqrt{\frac{2 kt}{m}}$
$For displacement x of the body, we have :$
$v = \frac{dx}{dr} = \sqrt{\frac{2 k}{m}} t^{1 / 2}$
$dx = k' t^{1 / 2} dt$
$Where k' = \sqrt{\frac{2 k}{3}} = New constant$

On integrating both sides, we get:

$\int d x = \int k' t^{1 / 2} d t$
$x = \frac{2}{3} k' t^{1 / 2}$
$\Rightarrow x \propto t^{3 / 2}$

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