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 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 cricket ball is thrown at a speed of \(28\) m/s in a direction \(30^{\circ}\) above the horizontal. The maximum height attained by the ball is:
1. \(5\) m
2. \(10\) m
3. \(15\) m
4. \(20\) m

An insect trapped in a circular groove of radius \(12\) cm moves along the groove steadily and completes \(7\) revolutions in \(100\) s. Is the acceleration vector a constant vector? What is its magnitude?
1. \(\mathrm{yes},\) \(2.3~\mathrm{cm/s^2}\)
2. \(\mathrm{no},\) \(5.3~\mathrm{cm/s^2}\)
3. \(\mathrm{yes},\) \(5.3~\mathrm{cm/s^2}\)
4. \(\mathrm{no},\) \(2.3~\mathrm{cm/s^2}\)

An insect trapped in a circular groove of radius \(12\) cm moves along the groove steadily and completes \(7\) revolutions in \(100\) s. What is the angular speed of the motion? 
1.\(0.62\) rad/s
2.\(0.06\) rad/s
3. \(4.40\) rad/s
4. \(0.44\) rad/s

A cricket ball is thrown at a speed of \(28\) m/s in a direction \(30^\circ\) above the horizontal. The time taken by the ball to return to the same level is:
1. \(2.5\) s
2.  \(2.9\) s
3.  \(3.5\) s
4.  \(3\) s

A hiker stands on the edge of a cliff \(490~\mathrm{m}\) above the ground and throws a stone horizontally with an initial speed of \(15\) m/s. Neglecting air resistance, the time taken by the stone to reach the ground and the speed with which it hits the ground are, respectively: (Take \(g=9.8~\mathrm{m/s^2}\))

Galileo, in his book Two new sciences, stated that for elevations that exceed or fall short of \(45^\circ\) by equal amounts, the ranges:

Rain is falling vertically with a speed of \(35\) m/s. A woman rides a bicycle with a speed of \(12\) m/s in the east-to-west direction. What is the direction in which she should hold her umbrella?

A particle starts from the origin at t = 0 with a velocity of 5.0î m/s and moves in the x-y plane under the action of a force that produces a constant acceleration of (3.0î +2.0ĵ ) $m/s^2$. What is the y-coordinate of the particle at the instant its x-coordinate is 84 m?

1. 36 m 

2.  26 m

3.  1 m

4.  0 m