Question

6.10 a body is moving unidirectionally under the influence of a source of constant power it's displacement in time t is proportional to ​

Answer 1

$\large\tt\purple{Solution⟹}$

We know that the power is given by:

$\sf \to \: p = fv$

$\to \sf \: mav = mv \times \frac{dv}{dt}$

$\to \sf \: k = constant$

$\to \sf \: vdv \: = \frac{k}{m} dt$

On integration :-

$\to \sf \: \frac{v {}^{2} }{2} = \frac{k}{m} dt$

$\to \sf \sqrt{ \frac{2kt}{m} }$

To get the displacement :-

$\to \sf \: v = \frac{dx}{dt} = \sqrt{ \frac{2k}{m} } t {}^{ \frac{1}{3} }$

$\sf \to \: dx = k {"}^{} \: t \frac{1}{2} \: dt$

$\sf \: where \: k {"}^{} = \sqrt{ \frac{2k}{3} }$

$\sf \to \: x = \frac{2}{3} \: k {"}^{} \: t {}^{ \frac{2}{3} }$

Hence, from the above equation it is shown that t is proportional to 2/3