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. \(\dfrac{\sqrt{3}}{2}\)

A bus is going to the North at a speed of \(30\) kmph. It makes a \(90^{\circ}\) left turn without changing the speed. The change in the velocity of the bus is:

Three particles are moving with constant velocities \(v_1 ,v_2\) and \(v\) respectively as given in the figure. After some time, if all the three particles are in the same line, then the relation among \(v_1 ,v_2\) and \(v\) is:
                            
1. \(v =v_1+v_2\)
2. \(v= \sqrt{v_{1} v_{2}}\)
3. \(v = \frac{v_{1} v_{2}}{v_{1} + v_{2}}\)
4. \(v=\frac{\sqrt{2} v_{1} v_{2}}{v_{1} + v_{2}}\)