A force $F=2t$ (where $t$) is time in seconds) is applied at $t=0sec$ to the block of mass $m$ placed on a rough horizontal surface. the coefficient of static and kinetic friction between the block and surface are ${\mu }_s$ and (\mu_k\) respectively. Which of the following graphs best representts the acceleraton vs time of the block $({\mu }_s>{\mu }_k)$?

Diagramme

The correct option is D Diagramme
Step1: Draw a diagram of given situation.

Diagrame

Step2: Find the graph acceleration vs time of the block.
Limiting static friction $f_1=\mu mg$
Suppose $t_0$ be the time when friction force is maximum,
$F=2t_0={\mu }_{s}mg$
The block just start moving imediatly after this instant with acceleration.
$a=\frac{{\mu }_{s}mg-{\mu }_{k}mg}{m}=({\mu }_s-{\mu }_k)g$
The acceleration of the block for $t>t_0$,
$a=\frac{2t-{\mu }_{k}mg}{m}$
If we compare this relation
$y=mx+c$
Slope of the graph,
$\tan \theta =\frac{2}{m}$
Therefore, graph is straight line

Final Answer : (a)