A particle moves along the positive branch of the curve \(y= \frac{x^{2}}{2}\) where \(x= \frac{t^{2}}{2}\), & \(x\) and \(y\) are measured in metres and in seconds respectively. At \(t= 2~\text{s}\), the velocity of the particle will be:
1. \(\left(\right. 2 \hat{i} - 4 \hat{j})~\text{m/s}\) 
2. \(\left(\right. 4 \hat{i} + 2 \hat{j}\left.\right)\text{m/s}\)
3. \(\left(\right. 2 \hat{i} + 4 \hat{j}\left.\right) \text{m/s}\)
4. \(\left(\right. 4 \hat{i} - 2 \hat{j}\left.\right) \text{m/s}\)

A deer wants to save her life from a lion. The lion follows a path whose equation is \(x^{2} + y^{2} = 16\). For saving her life, the deer needs to move on a path whose equation will be:
1. \(x^{2} + y^{2} = 4\)
2. \(x^{2} + y^{2} = 16\)
3. \(x^{2} + y^{2} - 64 = 0\)
4. Both (1) and (3) are correct.​​​​​​​

A particle rotates in a circular path starting from rest. If angular acceleration is \(4~\text{rad/s}^{2}\), then the time after which angle between net acceleration and tangential acceleration becomes \(45^{\circ}\) is?

A body is projected at an angle of \(30^{\circ}\) with the horizontal with a speed of \(30\) m/s. What is the angle made by the velocity vector with the horizontal after \(1.5\) sec? \(\left(g=10~\text{m/s}^2\right)\)
1. \(0^{\circ}\)
2. \(30^{\circ}\)
3. \(60^{\circ}\)
4. \(90^{\circ}\)

A cat is situated at point \(A\) (\(0,3,4\)) and a rat is situated at point \(B\) (\(5,3,-8\)$\mathrm{}$). The cat is free to move but the rat is always at rest. The minimum distance travelled by the cat to catch the rat is:
1. \(5\) unit
2. \(12\) unit
3. \(13\) unit
4. \(17\) unit

The position of a moving particle at time \(t\) is \(\overrightarrow{r}=3\hat{i}+4t^{2}\hat{j}-t^{3}\hat{k}.\) Its displacement during the time interval \(t=1\) s to \(t=3\) s will be:

An object moves at a constant speed along a circular path in a horizontal XY plane with its centre at the origin. When the object is at \(x=-2~\text{m}\), its velocity is \(-(4~\text{m/s})\hat j.\)$\stackrel{}{}$ What is the object's acceleration when it is at \(y= 2~\text{m}\)?
1. \(- 8~\text{m/s}^{2} \hat j\)
2. \(- 8~\text{m/s}^{2} \hat i\)
3. \(- 4~\text{m/s}^{2} \hat j\)
4. \(- 4~\text{m/s}^{2} \hat i\)

A particle has an initial velocity \(\overrightarrow{u} = \left(4 \hat{i} - 5 \hat{j}\right)\) m/s and it is moving with an acceleration \(\overrightarrow{a} = \left(\frac{1}{4} \hat{i} + \frac{1}{5} \hat{j}\right)\text{m/s}^{2}\). Velocity of the particle at \(t=2\) s will be:
1. \((6\hat i -4\hat j)~\text{m/s}\)
2. \((4.5\hat i -4.5\hat j)~\text{m/s}\)
3. \((4.5\hat i -4.6\hat j)~\text{m/s}\)
4. \((6\hat i -4.6\hat j)~\text{m/s}\)

A body is projected with a velocity of \(\left(3 \hat{i} + 4 \hat{j}\right)\text{m/s}\). The maximum height attained by the projectile is: (\(g=10\) ms^–2)

A particle of mass \(2\) kg is moving in a circular path with a constant speed of \(10\) m/s. The change in the magnitude of velocity when a particle travels from \(P\) to \(Q\) will be: [assume the radius of the circle is \(10/\pi^2]\)