First slide

Power

Question

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 :

Moderate

A

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

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

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

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

Solution

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

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