Question

A locomotive of mass m starts moving so that its velocity varies according to the law $b=k\sqrt{s}$ where k is constant and s is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first t seconds after the beginning of motion.

• A. $W=\frac{1}{8}m{k}^4{t}^2$
• B. $W=\frac{1}{4}{m}^2{k}^4{t}^2$
• C. $W=\frac{1}{4}m{k}^4{t}^4$
• D. $W=\frac{1}{8}m{k}^4{t}^4$

Answer 1

Answer: (A)

$W=\frac{1}{8}m{k}^4{t}^2$

$\frac{Ds}{dt}=ksqrt\left(s\right)$

$s^{\left(\frac{-1}{2}\right)}ds=kdt$

$\frac{s^{\left(\frac{-1}{2}\right)}}{\left(\frac{1}{2}\right)}=kt$2sqrt(s)=kt$

$s=\frac{k^2{t}^2}{4}$

$a=\frac{d^{2}s}{d{t}^2}=\frac{k^2}{2}$

$dW=FsdS=madS$

Since the force is constant,

$W=ma$ quad integral$\left(dS\right)=ma$ $quadS=\frac{m{k}^2}{2}\times \frac{k^2{t}^2}{4}$

$W=\frac{1}{8}m{k}^4{t}^2$