The coordinates of a moving particle at any time \(t\) are given by \(x= \alpha t^3\) and \(y = \beta t^3.\) The speed of the particle at a time \(t\) is given by:

1.	\(\sqrt{\alpha^{2} + \beta^{2}}\)	2.	\(3t \sqrt{\alpha^{2} + \beta^{2}}\)
3.	\(3t^{2} \sqrt{\alpha^{2} +\beta^{2}}\)	4.	\(t^{2} \sqrt{\alpha^{2} +\beta^{2}}\)

The figure shows a body of mass \(m\) moving with a uniform speed \(v\) along a circle of radius \(r\). The change in velocity in going from \(A\) to \(B\) is:

1. \(v\sqrt2\) $$

2. \(\dfrac v{\sqrt2}\)

3.  \(v\)

4. zero

A particle moves in a circular path with decreasing speed. Choose the correct statement. 

(1) Angular momentum remains constant

(2) Acceleration ($\overrightarrow{a}$) is towards the center

(3) Particle moves in a spiral path with decreasing radius

(4) The direction of angular momentum remains constant

The angle turned by a body undergoing circular motion depends on time as \(\theta=\theta_0+\theta_1t+\theta_2t^2.\) Then the angular acceleration of the body is:
1. \(\theta_1\)

2. \(\theta_2\)

3. \(2\theta_1\)

4. \(2\theta_2\)

A stone is just released from the window of a train moving along a horizontal, straight track. The stone will hit the ground following:

1. Straight path

2. Circular path

3. Parabolic path

4. Hyperbolic path

An aeroplane is flying at a constant horizontal velocity of \(600\ \text{km/hr}\) at an elevation of \(6\ \text{km}\) towards a point directly above the target on the Earth's surface. At an appropriate time, the pilot releases a ball so that it strikes the target at the Earth's surface. The ball will appear to be falling:

1. On a parabolic path as seen by the pilot in the plane

2. Vertically along a straight path as seen by an observer on the ground near the target

3. On a parabolic path as seen by an observer on the ground near the target

4. On a zig-zag path as seen by the pilot in the plane

A bomb is dropped from an aeroplane moving horizontally at constant speed. When air resistance is taken into consideration, the bomb:

1. Falls to earth exactly below the aeroplane

2. Fall to earth behind the aeroplane

3. Falls to earth ahead of the aeroplane

4. Flies with the aeroplane

An aeroplane is flying horizontally with a velocity \( u = 600\ \text{km/h}\) at a height of \(1960\ \text{m}\). When it is vertically at a point \(A\) on the ground, a bomb is released from it. The bomb strikes the ground at point \(B\). The distance \(AB\) is:

1. \(1200\ \text{m}\)

2. \(0.33\ \text{km}\)

3. \(3.33\ \text{km}\)

4.  \(33\ \text{km}\)

An aeroplane moving horizontally with a speed of \(720\ \text{km/h}\) drops a food packet while flying at a height of \(396.9\ \text{m}\). The time taken by a food packet to reach the ground and its horizontal range is:

1. \(3\ \text{s}\ \text{and}\ 2000\ \text{m}\)

2. \(5\ \text{s}\ \text{and}\ 500\ \text{m}\)

3. \(8\ \text{s}\ \text{and}\ 1500\ \text{m}\)

4. \(9\ \text{s}\ \text{and}\ 1800\ \text{m}\)

A particle \((A)\) is dropped from a height and another particle \((B)\) is thrown in the horizontal direction with a speed of \(5\ \text{m/s}\) from the same height. The correct statement is:

1. Both particles will reach at ground simultaneously

2. Both particles will reach at ground with same speed

3. Particle (A) will reach at ground first with respect to particle \((B)\)

4. Particle (B) will reach at ground first with respect to particle \((A)\)