Two buses $P$ and $Q$ start from a point at the same time and move in a straight line and their positions are represented by $X_{P}(t) = \alpha t + \beta t^{2}$ and $X_{Q}(t) = ft - t^{2}$. At what time,do both the buses have the same velocity?

$A$ man in a car at location $Q$ on a straight highway is moving with speed $v$. He decides to reach a point $P$ in a field at a distance $d$ from the highway (point $M$) as shown in the figure. The speed of the car in the field is half of that on the highway. What should be the distance $RM$ so that the time taken to reach $P$ is minimum?

At time $t=0$,a car moving along a straight line has a velocity of $16 \; m/s$. It slows down with an acceleration of $a = -0.5t \; m/s^2$,where $t$ is in seconds. Mark the correct statement$(s)$.

Which of the following statements are true for a moving body?

$A$ particle is constrained to move on a straight line path. It returns to the starting point after $10\, s$. The total distance covered by the particle during this time is $30\, m$. Which of the following statements about the motion of the particle is false?

Read each statement below carefully and state with reasons and examples,if it is true or false for a particle in one-dimensional motion:
$(a)$ With zero speed at an instant may have non-zero acceleration at that instant.
$(b)$ With zero speed may have non-zero velocity.
$(c)$ With constant speed must have zero acceleration.
$(d)$ With positive value of acceleration must be speeding up.