A particle of mass m moves on a straight line with its velocity increasing with distance according to the equation where is a constant. The total work done by all the forces applied on the particle during its displacement from to will be

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Solution

Q.
A particle of mass m moves on a straight line with its velocity increasing with distance according to the equation $v = α \sqrt{x},$ where $α$ is a constant. The total work done by all the forces applied on the particle during its displacement from $x = 0$ to $x = d,$ will be [2024]

1 $\frac{m}{2 α^{2} d}$

2 $\frac{m d}{2 α^{2}}$

3 $\frac{m α^{2} d}{2}$

4 $2 m α^{2} d$

Ans.
(3)

$v = α \sqrt{x}$

at $x = 0 : v = 0$ and at $x = d; v = α \sqrt{d}$

W.D. $= K_{f} - K_{i}$

W.D. $= \frac{1}{2} m {(α \sqrt{d})}^{2} - \frac{1}{2} m {(0)}^{2}$

$\Rightarrow$ W.D $= \frac{m α^{2} d}{2}$

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