A particle of mass m is driven by a machine that delivers a constant power k watts. If the particle starts from rest the force on the particle at time t is :

# A particle of mass m is driven by a machine that delivers a constant power k watts. If the particle starts from rest the force on the particle at time t is :

1. A

$\sqrt{\mathrm{mk}}{\mathrm{t}}^{-1/2}$

2. B

$\sqrt{2\mathrm{mk}}{\mathrm{t}}^{-1/2}$

3. C

$\frac{1}{2}\sqrt{\mathrm{mk}}{\mathrm{t}}^{-1/2}$

4. D

$\sqrt{\frac{\mathrm{mk}}{2}}{\mathrm{t}}^{-1/2}$

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### Solution:

P = Fv = mav
$⇒\mathrm{K}=\mathrm{mv}\frac{\mathrm{dv}}{\mathrm{dt}}$
By integrating the equation
$⇒\int \mathrm{vdv}=\int \frac{\mathrm{k}}{\mathrm{m}}\mathrm{dt}$
$⇒\frac{{\mathrm{v}}^{2}}{2}=\frac{\mathrm{k}}{\mathrm{m}}\mathrm{t}⇒\mathrm{v}=\sqrt{\frac{2\mathrm{k}}{\mathrm{m}}\mathrm{t}}$
$\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}}=\sqrt{\frac{2\mathrm{k}}{\mathrm{m}}}\left(\frac{1}{2}{\mathrm{t}}^{-1/2}\right)$
$\mathrm{F}=\mathrm{ma}=\mathrm{m}\left(\frac{1}{2}\right)\sqrt{\frac{2\mathrm{k}}{\mathrm{mt}}}⇒\mathrm{F}=\sqrt{\frac{\mathrm{mk}}{2\mathrm{t}}}$

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