A body of mass \(1~\text{kg}\) begins to move under the action of a time-dependent force \(\vec{F}=\left(2 t \hat{i}+3 t^2 \hat{j}\right) ~\text N,\) where \(\hat{i}\) and are unit vectors along the \({X}\) and \({Y}\text-\)axis. What power will be developed by the force at the time \((t)?\)

1. \(\left(2 t^2+4 t^4\right)~\text W\) 
2. \(\left(2 t^3+3 t^3\right) ~\text W\) 
3. \(\left(2 t^3+3 t^5\right) ~\text W\) 
4. \(\left(2 t^3+3 t^4\right) ~\text W\) 

A ball is thrown vertically downward from a height of \(20~\text m\) with an initial velocity \(v_0.\) It collides with the ground, loses \(50\%\) of its energy in a collision, and rebounds to the same height. The initial velocity \(v_0\) is: 
(Take, \(g=10~\text{ms}^{-2}\))
1. \(14~\text{ms}^{-1}\) 
2. \(20~\text{ms}^{-1}\)
3. \(28~\text{ms}^{-1}\)
4. \(10~\text{ms}^{-1}\)

Two similar springs \(P\) and \(Q\) have spring constants \(k_P\) and \(k_Q\), such that \(k_P>k_Q\). They are stretched, first by the same amount (case a), then by the same force (case b). The work done by the springs \(W_P\) and \(W_Q\) are related as, in case (a) and case (b), respectively:

Two particles of masses \(m_1\) and \(m_2\) move with initial velocities \(u_1\) and \(u_2\) respectively. On collision, one of the particles gets excited to a higher level, after absorbing energy \(E\). If the final velocities of particles are \(v_1\) and \(v_2\), then we must have:

A uniform force of \((3 \hat{i} + \hat{j})\) newton acts on a particle of mass \(2~\text{kg}.\) Hence the particle is displaced from the position \((2 \hat{i} + \hat{k})\) metre to the position \((4 \hat{i} + 3 \hat{j} - \hat{k})\) metre. The work done by the force on the particle is:
1. \(6~\text{J}\)
2. \(13~\text{J}\)
3. \(15~\text{J}\)
4. \(9~\text{J}\)

The potential energy of a system increases if work is done: