A ball is projected at a certain angle with initial velocity \(u\). It covers horizontal range \(R\). With what initial velocity it should be projected keeping the angle of projection the same so that its horizontal range becomes \(2.25R\)?

1.	\(2.5u\)	2.	\(1.5u\)
3.	\(2.25u\)	4.	\(0.25u\)

A particle is moving with velocity \(\overrightarrow{v} = k \left(y \hat{i} + x \hat{j}\right)\) where \(k\) is a constant. The general equation for the path will be:

A body is thrown vertically so as to reach its maximum height in \(t\) second. The total time from the time of projection to reach a point at half of its maximum height while returning (in second) is:
1. \(\sqrt{2} t\)
2. \(\left(1 + \frac{1}{\sqrt{2}}\right) t\)
3. \(\frac{3 t}{2}\)
4. \(\frac{t}{\sqrt{2}}\)

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. \(v=\frac{\sqrt{2} v_{1} v_{2}}{v_{1} + v_{2}}\)

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

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 \(1.0~\text{s}\), a particle goes from point \(A\) to point \(B\), moving in a semicircle of radius \(1.0~\text{m}\) (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. \(\theta_1\)
2. \(\theta_2\)
3. \(2\theta_1\)
4. \(2\theta_2\)

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: