A particle of mass m moves on a straight line with its velocity varying with the distance travelled according to the equation $v = α \sqrt{x}$ , where $α$ is a constant. Find the total work done by all the forces during a displacement from $x = 0$ to $x = b$ .

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$\frac{1}{2} α^{2} b$

$\frac{1}{2} m α^{2} b$

$\frac{1}{2} m α^{2}$

$\frac{1}{2} m^{2} α^{2} b$

answer is A.

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Detailed Solution

$v_{x = 0} = 0 : v_{x = b} = α \sqrt{b} \) ∴ W_{All} = Δ KE = K_{f} - K_{1} = \frac{1}{2} m (v_{f}^{2} - v_{j}^{2}) = \frac{1}{2} \times m [(α \sqrt{b})^{2} - 0] = \frac{1}{2} m α^{2} b$

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