Question

Mark the correct statements for a particle going on a straight line:
(a) If the velocity and acceleration have opposite sign, the object is slowing down.
(b) If the position and velocity have opposite sign the particle is moving towards the origin.
(c) If the velocity is zero at an instant, the acceleration should also be zero at that instant.
(d) If the velocity is zero for a time interval, the acceleration is zero at any instant within the time interval.

Answer 1

(a) If the velocity and acceleration have opposite signs, the object slows down.
(b) If the position and velocity have opposite signs, the particle moves towards the origin.
(d) If the velocity is zero for a time interval, the acceleration is zero at any instant within the time interval.

(a) Acceleration is given by
$$\begin{gathered} -a=\frac{dv}{dt} \\ -a<0 \\ \Rightarrow \frac{dv}{dt}<0 \\ \Rightarrow V_{\mathrm{final}}<V_{\mathrm{initial}} \end{gathered}$$

(b) If the position and velocity have opposite signs, the particle moves towards the origin. It can be explained by following figure:

(c) If the velocity is zero in a certain time interval, then the change in the velocity in that time interval will also be zero. As acceleration is rate of change of velocity, it will also be zero at an instant in that time interval.