A block is placed on a rough horizontal plane. A time dependent horizontal force $$F = kt$$ acts on the block, where k is a positive constant. The acceleration - time graph of the block is :

Initially, the applied force increases from zero ($$F = kt$$). The block remains stationary as long as the force is less than or equal to the maximum limiting static friction ($$f_{s,max} = \mu_s mg$$). Acceleration: $$a = 0$$ for $$t \leq \frac{\mu_s mg}{k}$$.

Once motion begins, the friction coefficient suddenly drops from static ($$\mu_s$$) to kinetic ($$\mu_k$$, where $$\mu_k < \mu_s$$). This sudden reduction in friction causes an instantaneous net force, forcing the acceleration to jump vertically from zero to a finite value.

Once the block is moving, kinetic friction remains constant ($$f_k = \mu_k mg$$). The equation of motion becomes:

$$kt - \mu_k mg = ma \implies a = \left(\frac{k}{m}\right)t - \mu_k g$$

Since this follows a linear form ($$y = mx + c$$), the acceleration increases along a straight line with a positive slope.

The correct graph showing a flat line at zero, a vertical jump, and a linear incline is Option B.