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

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 $\theta ={\theta }_0+{\theta }_{1}t+{\theta }_2{t}^2$. Then the angular acceleration of the body is 

(1) θ₁

(2) θ₂

(3) 2θ₁

(4) 2θ₂

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 km/hr at an elevation of 6 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. The ball will appear to be falling

(1) On a parabolic path as seen by 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 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 km/h at a height of 1960 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 m

2. 0.33 km

3. 3.33 km

4. 33 km

An aeroplane moving horizontally with a speed of 720 km/h drops a food pocket while flying at a height of 396.9 m. The time taken by a food pocket to reach the ground and its horizontal range is (Take g = 9.8 m/sec²)

(1) 3 sec and 2000 m

(2) 5 sec and 500 m

(3) 8 sec and 1500 m

(4) 9 sec and 1800 m

A particle (A) is dropped from a height and another particle (B) is thrown in the horizontal direction with a speed of 5 m/sec 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)

A bomber plane moves horizontally with a speed of \(500~\text{m/s}\) and a bomb is released from it. The bomb strikes the ground in \(10~\text{s}\). The angle at which it strikes the ground will be: \((g= 10~\text{m/s}^2)\)
1. \(\tan ^{-1}\left(\frac{1}{5}\right )\)
2. \(\tan \left(\frac{1}{5}\right)\)
3. \(\tan ^{-1}(1)\)
4. \(\tan ^{-1}(5)\)