If the equation for the displacement of a particle moving on a circular path is given by \(\theta = 2t^3 + 0.5\) where θ is in radians and t in seconds, then the angular velocity of the particle after 2 sec from its start is:

1. 8 rad/sec

2. 12 rad/sec

3. 24 rad/sec

4. 36 rad/sec

The coordinates of a moving particle at any time ‘t’ are given by x = αt³ and y = βt³. The speed of the particle at time ‘t’ is given by:

1. $\sqrt{{\alpha }^2+{\beta }^2}$

2. $3$ $t\sqrt{{\alpha }^2+{\beta }^2}$

3. $3$ $t^2\sqrt{{\alpha }^2+{\beta }^2}$

4. $t^2\sqrt{{\alpha }^2+{\beta }^2}$

At the top of the trajectory of a projectile, the directions of its velocity and acceleration are:

Four bodies P, Q, R and S are projected with equal velocities having angles of projection 15^o, 30^o, 45^o and 60^o with the horizontal respectively. The body having the shortest range is? 

A stone projected with a velocity u at an angle θ with the horizontal reaches maximum height H₁. When it is projected with velocity u at an angle $\left(\frac{\pi }{2}-\theta \right)$with the horizontal, it reaches maximum height H₂. The relation between the horizontal range of the projectile R and H₁ & H₂ is: 

A cricketer can throw a ball to a maximum horizontal distance of 100 m. With the same effort, he throws the ball vertically upwards. The maximum height attained by the ball is 

1. 100 m

2. 80 m

3. 60 m

4. 50 m

The horizontal range of a projectile is $4\sqrt{3}$ times its maximum height. Its angle of projection will be:

1. 45°

2. 60°

3. 90°

4. 30°

A boat is moving with a velocity 3i + 4j with respect to ground. The water in the river is moving with a velocity – 3i – 4j with respect to ground. The relative velocity of the boat with respect to water is 

1. 8j

2. – 6i – 8j

3. 6i +8j

4. $5\sqrt{2}$

A river is flowing from east to west at a speed of 5 m/min. A man on south bank of river, capable of swimming 10m/min in still water, wants to swim across the river in shortest time. He should swim:

1. Due north

2. Due north-east

3. Due north-east with double the speed of the river

4. None of the above

A man sitting in a bus travelling in a direction from west to east with a speed of 40 km/h observes that the rain-drops are falling vertically downwards. To another man standing on ground the rain will appear 

1. To fall vertically downwards

2. To fall at an angle going from west to east

3. To fall at an angle going from east to west

4. The information given is insufficient to decide the direction of the rain