$A$ car is moving along a straight line,say $OP$ in the given figure. It moves from $O$ to $P$ in $18\; s$ and returns from $P$ to $Q$ in $6.0\; s$. What are the average velocity and average speed of the car in going from $O$ to $P$ and back to $Q?$

State whether the following statements are true or false:
$(a)$ Velocity can change without a change in speed.
$(b)$ The total distance covered by a freely falling body in a given time is directly proportional to the time.
$(c)$ $A$ particle is a point object that possesses dimensions.

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.

Two trains '$A$' and '$B$' of length '$l$' and '$4l$' are travelling into a tunnel of length '$L$' on parallel tracks from opposite directions with velocities $108\,km/h$ and $72\,km/h$,respectively. If train '$A$' takes $35\,s$ less time than train '$B$' to cross the tunnel,then the length '$L$' of the tunnel is $...........\,m$. (Given $L = 60l$)

$A$ particle moves along a straight line such that its displacement $x$ varies with time $t$ as $x = \alpha t^3 + \beta t^2 + \gamma$, where $\alpha, \beta, \gamma$ are constants. $V_1$ is the average velocity of the particle during its journey between $t = 1 \ s$ and $t = 3 \ s$. $V_2$ is the instantaneous velocity of the particle at $t = 3 \ s$. The ratio $\frac{V_1}{V_2}$ is

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$(s)$.