A particle moving in a circle of radius \(R\) with a uniform speed takes a time \(T\) to complete one revolution. If this particle were projected with the same speed at an angle \(\theta\) to the horizontal, the maximum height attained by it equals \(4R.\) The angle of projection, \(\theta\) is then given by:
1. \( \theta=\sin ^{-1}\left(\frac{\pi^2 {R}}{{gT}^2}\right)^{1/2}\)
2. \(\theta=\sin ^{-1}\left(\frac{2 {gT}^2}{\pi^2 {R}}\right)^{1 / 2}\)
3. \(\theta=\cos ^{-1}\left(\frac{{gT}^2}{\pi^2 {R}}\right)^{1 / 2}\)
4. \(\theta=\cos ^{-1}\left(\frac{\pi^2 {R}}{{gT}^2}\right)^{1 / 2}\)

A car starts from rest and accelerates at $5$ $\mathrm{m}/{\mathrm{s}}^2$. At \(t=4\) s, a ball is dropped out of a window by a person sitting in the car. What is the velocity and acceleration of the ball at \(t=6\) s? (Take \(g=10\) m/s²)
1. $20\sqrt{2}$ $\mathrm{m}/\mathrm{s}, \mathrm{0 }\mathrm{m}/{\mathrm{s}}^2$
2. $20\sqrt{2}$ $\mathrm{m}/\mathrm{s},$ $10$ $\mathrm{m}/{\mathrm{s}}^2$
3. $20$ $\mathrm{m}/\mathrm{s},$ $5$ $\mathrm{m}/{\mathrm{s}}^2$
4. $20$ $\mathrm{m}/\mathrm{s},$ $0$

A particle moves along a circle of radius \({{20}\over{\pi}} m\) with constant tangential acceleration. If the velocity of the particle is 80 m/s at the end of the second revolution after motion has begun, the tangential acceleration is:

1. 40 ms^–2

2. 640π ms^–2

3. 160π ms^–2

4. 40π ms^–2

Two particles having mass \(M\) and \(m\) are moving in a circular path having radius \(R\) & \(r\) respectively. If their time periods are the same, then the ratio of angular velocities will be: 
1. \(\frac{r}{R}\)
 2. \(\frac{R}{r}\)
3. \(1\)
4. \(\sqrt{\frac{R}{r}}\)

A particle is projected, making an angle of $\mathrm{}$\(45^{\circ}\) with the horizontal and having kinetic energy \(K\). The kinetic energy at the highest point will be: 
1. \(\frac{K}{\sqrt{2}}\)
2. \(\frac{K}{2}\)
3. \(2K\)
4. \(K\)

A particle (A) is dropped from a height and another particle (B) is projected in a horizontal direction with a speed of 5 m/s from the same height. The correct statement, from the following, is:

Two particles are projected with the same initial velocity, one makes an angle θ with the horizontal while the other makes an angle θ with the vertical. If their common range is R, then the product of their time of flight is directly proportional to:
1.  R
2.  $R^2$
3.  $\frac{1}{R}$
4.  $R^0$

Two particles are separated by a horizontal distance x as shown in the figure. They are projected at the same time as shown in the figure with different initial speeds. The time after which the horizontal distance between them becomes zero will be:
                     
1.  $\frac{x}{u}$

2.  $\frac{u}{2x}$

3.  $\frac{2u}{x}$

4.  None of the above

Two boys are standing at the ends \(A\) and \(B\) of the ground where \(AB =a.\) The boy at \(B\) starts running in a direction perpendicular to \(AB\) with velocity \(v_1.\) The boy at \(A\) starts running simultaneously with velocity \(v\) and catches the other boy in a time \(t,\) where \(t\) is:

A stone tied to the end of a 1 m long string is whirled in a horizontal circle at a constant speed. If the stone makes 22 revolutions in 44 seconds, what is the magnitude and direction of acceleration of the stone?