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:

• A. $\sqrt{\frac{m k}{2}} \, t^{- \frac{1}{2}}$
• B. $\sqrt{m k} \, t^{- \frac{1}{2}}$
• C. $\sqrt{2 m k} \, t^{- \frac{1}{2}}$
• D. $\frac{1}{2}\sqrt{m k} \, t^{- \frac{1}{2}}$

Answer 1

Answer: (A)

$Fv=constant=k$
$\frac{m d v}{d t}v=k$
$\displaystyle \int v d v=\displaystyle \int \frac{k}{m}dt \, $
$\frac{v^{2}}{2}=\frac{k}{m}t$
$v=\sqrt{\frac{2 k}{m} t}$
$F=\frac{m d v}{d t}$
$=m \, \sqrt{\frac{2 k}{m}}\frac{1}{2}t^{- \frac{1}{2}} \, $
$=\sqrt{\frac{m k}{2}} \, t^{- \frac{1}{2}}$