Question

A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v=a\sqrt{s}$ where $a$ is a 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.

Answer 1

Step 1: Given

Mass, $m$

Velocity, $v=a\sqrt{s}$; here a is a constant, and s is the distance covered.

Step 2: Formula used

$$\begin{gathered} v=\frac{ds}{dt} \\ A=v\frac{dv}{ds} \\ W=m\times A\times s \end{gathered}$$

Where, “W” is work done.

Step 3: Find the value of $s$

The velocity is given as $v=a\sqrt{s}$

$\frac{ds}{dt}=a\sqrt{s}$

Integrating distance from $0$ to $x$ and time from $0$ to $t$.

$$\begin{gathered} {\int }_0^x\frac{ds}{a\sqrt{s}}={\int }_0^{t}dt \\ \Rightarrow \frac{2\sqrt{s}}{a}=t \\ \Rightarrow s=\frac{a^2{t}^2}{4} \end{gathered}$$

Step 4: Find acceleration, $A$

$$\begin{gathered} A=v\frac{dv}{ds} \\ \Rightarrow A=a\sqrt{s}\times \frac{a}{2\sqrt{s}} \\ \Rightarrow A=\frac{a^2}{2} \end{gathered}$$

Step 5: Find the total work done;

$$\begin{gathered} W=m\times A\times s \\ \Rightarrow W=m\times \frac{a^2}{2}\times \frac{a^2{t}^2}{4} \\ \Rightarrow W=\frac{m{a}^4{t}^2}{8} \end{gathered}$$

Thus, the total work done by all forces acting on the locomotive is $\frac{m{a}^4{t}^2}{8}$