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

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 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 particle moves with constant speed v along a circular path of radius r and completes the circle in time T. The acceleration of the particle is:

1. $2$ $\pi$ $v/T$

2. $2$ $\pi$ $r/T$

3. $2$ $\pi$ $r^2/T$

4. $2$ $\pi$ $v^2/T$

What is the value of linear velocity if \(\overset{\rightarrow}{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}\) and \(\overset{\rightarrow}{r} = 5\hat{i} - 6\hat{j} + 6\hat{k}\)  :

The time taken by a block of wood (initially at rest) to slide down a smooth inclined plane 9.8 m long (angle of inclination is $30°$) is:

An aeroplane is moving with a velocity \(u\). It drops a packet from a height \(h\). The time \(t\) taken by the packet to reach the ground will be:
1. \( \sqrt{\left(\frac{2 g}{h}\right)} \)
2. \( \sqrt{\left(\frac{2 u}{g}\right)} \)
3. \( \sqrt{\left(\frac{h}{2 g}\right)} \)
4. \( \sqrt{\left(\frac{2 h}{g}\right)}\)

A particle moves so that its position vector is given by \(r=\cos \omega t \hat{x}+\sin \omega t \hat{y}\) where \(\omega\) is a constant.  Based on the information given, which of the following is true?

Two men P & Q are standing at corners A & B of square ABCD of side 8 m. They start moving along the track with constant speed 2 m/s and 10 m/s respectively. The time when they will meet for the first time, is equal to:
     
1. 2 sec

2. 3 sec

3. 1 sec

4. 6 sec

The velocity of a projectile at the initial point \(A\) is \(2\hat i+3\hat j~\)m/s. Its velocity (in m/s) at point \(B\) is: