Question

Statement -1 : Let $v$ be the speed of particle at any time $t$. If $\frac{dv}{dt}$ is a non - zero constant then speed of particle may be minimum at some time.
Statement -2 : The speed $v$ of a moving particle may be minimum at time $t=t_0$ when $v=0$. The speed $v$ of a moving particle may also be minimum at time $t=t_0$ when $\frac{dv}{dt}=0$ and $\frac{d^{2}v}{d{t}^2}=$ positive.

• A. Both statement 1 and statement 2 are correct and statement 2 is the correct explanation of statement 1.
• B. Both statement 1 and statement 2 are correct but statement 2 is not the correct explanation of statement 1.
• C. Statement 1 is correct and statement 2 is incorrect.
• D. Statement 1 is incorrect and statement 2 is correct.

Answer 1

The correct option is B Both statement 1 and statement 2 are correct but statement 2 is not the correct explanation of statement 1.

For a stone projected vertically upwards, the speed $v$ is least at highest point even though $\frac{dv}{dt}$ is constant. Hence statement 1 is true.
Suppose the velocity of a particle varies with time as $v=t^3-t^2-t+1$. Here the velocity is minimum at $t=1\text{s}$ as $v=0$ at $t=1\text{s}$. If we differentiate v w.r.t $t$, we get $\frac{dv}{dt}=3t^2-2t-1$.
Here $\frac{dv}{dt}=0$, at $t=1\text{s}$ and $\frac{d^{2}v}{d{t}^2}=6t-2=6\left(1\right)-2=4=$ positive.
Hence, statement 2 is also true but it is not the correct explanation of statement 1.