A particle starting from the origin \((0,0)\) moves in a straight line in the \((x,y)\) plane. Its coordinates at a later time are ($\sqrt{3}$, \(3).\) The path of the particle makes an angle of __________ with the \(x\)-axis:
1. \(30^\circ\)
2. \(45^\circ\)
3. \(60^\circ\)
4. \(0\)

For a projectile projected at angles \((45^{\circ}-\theta)\) and \((45^{\circ}+\theta)\), the horizontal ranges described by the projectile are in the ratio of:
1. \(1:1\)
2. \(2:3\)
3. \(1:2\)
4. \(2:1\)

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:

Two particles \(A\) and \(B\) are moving in a uniform circular motion in concentric circles of radii \(r_A\) and \(r_B\) with speeds \(v_A\) and \(v_B\) respectively. Their time periods of rotation are the same. The ratio of the angular speed of \(A\) to that of \(B\) will be:

A particle is projected with a speed \(u\) at an angle \(\theta\) to the horizontal. Radius of curvature at highest point of its trajectory is?
1. \(\frac{u^{2} \cos^{2} \theta}{2 g}\)
2. \(\frac{\sqrt{3} u^{2}\cos^{2} \theta}{2 g}\)
3. \(\frac{u^{2} \cos^{2} \theta}{g}\)
4. \(\frac{\sqrt{3} u^{2} \cos^{2} \theta}{g}\)

Figure below shows a body of mass \(M\) moving with a uniform speed \(v\) on a circular path of radius, \(R\). What is the change in acceleration in going from \(P_1\) to \(P_2\)? 
                    

1. zero
2. \(v^{2} / 2 R\)
3. \(2 v^{2} / R\)
4. \(\frac{v^{2}}{R} \times \sqrt{2}\)

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 particle moves along a parabolic path \(y =9x^2\) in such a way that the \(x\) component of the velocity remains constant and has a value of \(\frac{1}{3}~\text{m/s}\)${\mathrm{}}^{}$. It can be deduced that the acceleration of the particle will be:
1. \(\frac{1}{3}\hat j~\text{m/s}^2\)
2. \(3\hat j~\text{m/s}^2\)
3. \(\frac{2}{3}\hat j~\text{m/s}^2\)
4. \(2\hat j~\text{m/s}^2\)

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:

A missile is fired for a maximum range with an initial velocity of \(20~\text {m/s}.\) If \(g=10~\text{m/s}^2,\) then the range of the missile will be: