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

A particle starts from the origin at t=0 and moves in the x-y plane with a constant acceleration 'a' in the y direction. Its equation of motion is $y=b{x}^2$. The x component of its velocity (at t=0) will be:

1. variable

2. $\sqrt{\frac{2a}{b}}$

3. $\frac{a}{2b}$

4. $\sqrt{\frac{a}{2b}}$

A boat moves with a speed of 5 km/h relative to water in a river flowing with a speed of 3 km/h. Width of the river is 1 km. The minimum time taken for a round trip will be

1. 5 min

2. 60 min

3. 20 min

4. 30 min

A river is flowing from W to E with a speed of 5 m/min. A man can swim in still water with a velocity of 10 m/min. In which direction should the man swim so as to take the shortest possible path to go to the south. 

1. 30° with downstream

2. 60° with downstream

3. 120° with downstream

4. South

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:

In 1.0 s, a particle goes from point A to point B, moving in a semicircle of radius 1.0 m (see figure). The magnitude of the average velocity is:
                   

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 particle is moving eastwards with velocity of 5 m/s. In 10 seconds the velocity changes to 5 m/s northwards. The average acceleration in this time is?

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: