The speed of a boat is \(5\) km/hr in still water. It crosses a river of width \(1\) km along the shortest possible path in \(15\) minutes. The velocity of the river water is:
1. \(3\) km/hr
2. \(4\) km/hr
3. \(5\) km/hr
4. \(2\) km/hr

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:
 

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

If two projectiles, with the same masses and with the same velocities, are thrown at an angle \(60^\circ\) and \(30^\circ\) with the horizontal, then which of the following quantities will remain the same?

The width of the river is \(1\) km. The velocity of the boat is \(5\) km/hr. The boat covered the width of the river with the shortest possible path in \(15\) min. Then the velocity of the river stream is:
1. \(3\) km/hr
2. \(4\) km/hr
3. \(\sqrt{29}\) km/hr
4.  \(\sqrt{41}\) km/hr

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. \(K\)

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 \((A)\) 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:

A particle moves along a circle of radius \(\frac{20}{\pi}~\text{m}\) 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. \(640\pi\) ms^–2

3. \(160\pi\) ms^–2

4. \(40\pi\) ms^–2

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?