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:

1.	\(\vec{r_{1  }} . \vec{v_{1}}   = \vec{r_{2  }} . \vec{v_{2}}\)	2.	\(\vec{r_{1}} \times\vec{v_{1}}   = \vec{r_{2}} \times \vec {v_{2}}\)
3.	\(\vec{r_{1}}-\vec{r_{2}}=\vec{v_{1}} - \vec{v_{2}}\)	4.	\(\frac{\vec{r_{1}} - \vec{r_{2}}}{\left|\vec{r_{1}} -  \vec{r_{2}}\right|} =   \frac{\vec{v_{2}} -  \vec{v_{1}}}{\left|\vec{v_{2}} -   \vec{v_{1}}\right|}\)

Three balls are thrown from the top of a building with equal speeds at different angles. When the balls strike the ground, their speeds are \(v_{1} , v_{2}\) \(\text{and}\) \(v_{3}\) respectively, then:
              

1. \(v_{1} > v_{2} > v_{3}\)
2. \(v_{3} > v_{2} = v_{1}\)
3. \(v_{1} = v_{2} = v_{3}\)
4. \(v_{1} < v_{2} < v_{3}\)${\mathrm{}}_{}$

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:

A body is moving with a velocity of \(30~\text{m/s}\) towards the east. After \(10~\text s,\) its velocity becomes \(40~\text{m/s}\) towards the north. The average acceleration of the body is:
1. \( 7~\text{m/s}^2\)
2. \( \sqrt{7}~\text{m/s}^2\)
3. \(5~\text{m/s}^2\)
4. \(1~\text{m/s}^2\)

The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\text{m/s}.\) Its velocity (in m/s) at the point \(B\) 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. \(v=\frac{\sqrt{2} v_{1} v_{2}}{v_{1} + v_{2}}\)

Certain neutron stars are believed to be rotating at about \(1\) rev/s. If such a star has a radius of \(20\) km, the acceleration of an object on the equator of the star will be:

The angle turned by a body undergoing circular motion depends on the time as given by the equation, \(\theta = \theta_{0} + \theta_{1} t + \theta_{2} t^{2}\). It can be deduced that the angular acceleration of the body is? 
1. \(\theta_1\)
2. \(\theta_2\)
3. \(2\theta_1\)
4. \(2\theta_2\)

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: