Consider the motion of the tip of the second hand of a clock. In one minute (assuming $R$ to be the length of the second hand), its:

Three girls skating on a circular ice ground of radius $200$ m start from a point $P$ on the edge of the ground and reach a point $Q$ diametrically opposite to $P$ following different paths as shown in the figure. The correct relationship among the magnitude of the displacement vector for three girls will be:

1. $A > B > C$
2. $C > A > B$
3. $B > A > C$
4. $A = B = C$

A cat is situated at point $A$ ($0,3,4$) and a rat is situated at point $B$ ($5,3,-8$). The cat is free to move but the rat is always at rest. The minimum distance travelled by the cat to catch the rat is:
1. $5$ unit
2. $12$ unit
3. $13$ unit
4. $17$ unit

A particle is moving on a circular path of radius $R.$ When the particle moves from point $A$ to $B$ (angle $\theta$), the ratio of the distance to that of the magnitude of the displacement will be:

         
1. $\dfrac{\theta}{\sin\frac{\theta}{2}}$
2. $\dfrac{\theta}{2\sin\frac{\theta}{2}}$
3. $\dfrac{\theta}{2\cos\frac{\theta}{2}}$
4. $\dfrac{\theta}{\cos\frac{\theta}{2}}$

A particle is moving such that its position coordinates $(x,y)$ are $​ (2~\text m,  3~\text m)​$ at time $t=0,$   $​ (6~\text m,  7~\text m)​$ at time $t=2~\text s$  and $​ (13~\text m,  14~\text m)​$ at time $t=5~\text s.$  The average velocity vector $(v_{avg})$ from $t=0$ to $t=5~\text s$ is:

A car turns at a constant speed on a circular track of radius $100~\text m,$ taking $62.8~\text s$ for every circular lap. The average velocity and average speed for each circular lap, respectively, is:

The coordinates of a moving particle at any time $t$ are given by $x= \alpha t^3$ and $y = \beta t^3.$ The speed of the particle at a time $t$ is given by:

Two particles $A$ and $B,$ move with constant velocities $\vec{v_1}$ and $\vec{v_2}.$ At the initial moment their position vector are $\vec {r_1}$ and $\vec {r_2}$ respectively. The conditions for particles $A$ and $B$ for their collision to happen will be:

Two particles move from $A$ to $C$ and $A$ to $D$ on a circle of radius $R$ and the diameter $AB.$ If the time taken by both particles is the same, then the ratio of magnitudes of their average velocities is:
                                
1. $2$
2. $2\sqrt{3}$
3. $\sqrt{3}$
4. $\dfrac{\sqrt{3}}{2}$