1. Use the work–energy theorem

The total work done by all forces is equal to the change in kinetic energy:

$W = \Delta K = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$

2. Given velocity as a function of position

We are given:

$v = \alpha \sqrt{x}$

So,

$v^2 = \alpha^2 x$

3. Find initial and final velocities

• At $x=0$: $v_i = \alpha \sqrt{0} = 0$

• At $x=d$: $v_f = \alpha \sqrt{d}$ so $v_f^2 = \alpha^2 d$

4. Compute the work done

$W = \frac{1}{2}m(\alpha^2 d) - \frac{1}{2}m(0)$

$W = \frac{1}{2} m \alpha^2 d$

5. Match with the options

$W = \frac{m\alpha^2 d}{2}$

This corresponds to Option C.