Q.Power is the rate of doing work or the rate at which energy is supplied to the system. A constant force F is applied to a body of mass m, starting from rest. Power delivered by the force at time t from the start is proportional to

[!TLDR] $P=Fv=F(at)=F\left(\frac{F}{m}\right)t=\frac{F^2t}{m}$, so power is directly proportional to t; option (a). [!ANSWE …

Starting from rest under a constant force F, the acceleration is $a=\frac{F}{m}$ (constant), so the speed at time t is $v=at=\frac{Ft}{m}$. Power delivered by the force at time t is $P=\vec{F}\cdot\vec{v}=Fv$ (force and velocity are along the same line here, since the force is constant and the body starts from rest), so $P=F\left(\frac{Ft}{m}\right)=\frac{F^2t}{m}$. Since $\frac{F^2}{m}$ is a constant (for a given F and m), this shows $P\propto t$ directly -- i.e. power delivered by a constant force, starting from rest, increase …

Derive $v=at=(F/m)t$ from the constant force and starting-from-rest condition, then compute $P=Fv$ and …

Confusing power (rate of doing work, $P=Fv$, linear in t here) with WORK ITSELF ($W=\frac{1}{2}mv^2=\frac{F^2t^2}{2m}$, which is proportional to $t^2$) -- the …