Question

The speed reached by a car of mass ${}^{\prime }{m}^{\prime }$ in travelling a distance ${}^{\prime }{x}^{\prime }$, driven by constant power ${}^{\prime }{p}^{\prime }$ is given by :

• A. $v=\frac{3xP}{m}$
• B. $v={\left(\frac{3xP}{m}\right)}^{1/2}$
• C. $v={\left(\frac{3xP}{m}\right)}^{1/3}$
• D. $v={\left(\frac{3xP}{m}\right)}^2$

Answer 1

Answer: (C)

$v={\left(\frac{3xP}{m}\right)}^{1/3}$

Power $=P=fv$ ---- $\left(i\right)$
$f=me$ ---- $\left(ii\right)$
From $\left(i\right)$ & $\left(ii\right)$
$P=mev$ ---- $\left(iii\right)$
$a=v\frac{dv}{dx}$ ---- $\left(iv\right)$
From $\left(iii\right)$ & $\left(iv\right)$
$P=m.\left(v\frac{dv}{dx}\right)v$
$P=m{v}^2\frac{dv}{dx}$
$\frac{P}{m}{\int }_0^{x}dx={\int }_0^v\frac{v^2}{}dv$
$\frac{Px}{m}=\frac{v^3}{3}$
$v^3=\frac{3Px}{m}$
$v={\left(\frac{3Px}{m}\right)}^{1/3}$