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 condition for particles A and B for their collision to happen will be:

1.  $\overrightarrow{{\mathrm{r}}_{1 }}.\overrightarrow{{\mathrm{v}}_1} =\overrightarrow{{\mathrm{r}}_{2 }}.\overrightarrow{{\mathrm{v}}_2}$

2.  $\overrightarrow{{\mathrm{r}}_1} \mathrm{x} \overrightarrow{{\mathrm{v}}_1} =\overrightarrow{{\mathrm{r}}_2} \mathrm{x} \overrightarrow{{\mathrm{v}}_2}$

3. $\overrightarrow{{\mathrm{r}}_1}- \stackrel{\rightharpoonup }{{\mathrm{r}}_2} =\overrightarrow{{\mathrm{v}}_1}- \overrightarrow{{\mathrm{v}}_2}$

4. $\frac{\overrightarrow{{\mathrm{r}}_1}- \overrightarrow{{\mathrm{r}}_2}}{\left|\overrightarrow{{\mathrm{r}}_1}- \overrightarrow{{\mathrm{r}}_2}\right|}= \frac{\overrightarrow{{\mathrm{v}}_2}- \overrightarrow{{\mathrm{v}}_1}}{\left|\overrightarrow{{\mathrm{v}}_2}- \overrightarrow{{\mathrm{v}}_1}\right|}$

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 ${\mathrm{v}}_1,{\mathrm{v}}_2$ $\mathrm{and}$ ${\mathrm{v}}_3$ respectively, then:
           

1. ${\mathrm{v}}_1>{\mathrm{v}}_2>{\mathrm{v}}_3$

2. ${\mathrm{v}}_3>{\mathrm{v}}_2={\mathrm{v}}_1$

3. ${\mathrm{v}}_1={\mathrm{v}}_2={\mathrm{v}}_3$

4. ${\mathrm{v}}_1<{\mathrm{v}}_2<{\mathrm{v}}_3$

A car turns at a constant speed on a circular track of radius \(100\) m, taking \(62.8\) 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 \(30\) m/s towards the east. After \(10\) s, its velocity becomes \(40\) m/s towards the north. The average acceleration of the body is:
1. \( 7 \mathrm{~m} / \mathrm{s}^2 \)
2. \( \sqrt{7} \mathrm{~m} / \mathrm{s}^2 \)
3. \( 5 \mathrm{~m} / \mathrm{s}^2 \)
4. \( 1 \mathrm{~m} / \mathrm{s}^2\)

The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\)m/s. Its velocity (in m/s) at point \(B\) is:
              

Three particles are moving with constant velocities ${\mathrm{v}}_1,$ ${\mathrm{v}}_2$ 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 ${\mathrm{v}}_1,$ ${\mathrm{v}}_2$ and v is:
                                    
1.  $\mathrm{v}=$ ${\mathrm{v}}_1$ $+$ ${\mathrm{v}}_2$

2.  $\mathrm{v}=$ $\sqrt{{\mathrm{v}}_1{\mathrm{v}}_2}$

3.  $\mathrm{v}=\frac{{\mathrm{v}}_1{\mathrm{v}}_2}{{\mathrm{v}}_1+{\mathrm{v}}_2}$

4.  $\mathrm{v}=$ $\frac{\sqrt{2}{\mathrm{v}}_1{\mathrm{v}}_2}{{\mathrm{v}}_1+{\mathrm{v}}_2}$

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. θ₁

2. θ₂

3. 2θ₁

4. 2θ₂

A vector $\overrightarrow{a}$ is turned without a change in its length through a small angle $d\theta .$ The value of $\left|\Delta \overrightarrow{a}\right|$ and $\Delta a$ are, respectively:

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