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 done by all the forces acting on the locomotive during the first t seconds after the beginning of the motion.

Answer 1

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Work done = Force . Displacement
=> W = f.s

Velocity is given
$v = a \sqrt{s}$
where a is constant and s is the distance covered

We have,
Force = mass . acceleration
$f = mA$

And
Acceleration = Rate of change of Velocity
$A = \frac{d}{dt}(v)$

=> $A = \frac{d}{dt}(a \sqrt{s})$

we have
$\frac{d}{dt}(\sqrt{x})= \frac{1}{2\sqrt{x}}$

So
=> $A = a\frac{1}{2\sqrt{s}}\frac{ds}{dt}$

We know
velocity $v = \frac{ds}{dt} = a \sqrt{s}$

=> $A = a\frac{1}{2\sqrt{s}}(a\sqrt{s})$
=> $A= \frac{a^{2}}{2}$

Therefore
$f = m \frac{a^{2}}{2}$

$w= m \frac{a^{2}}{2}s$