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

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

A ball is thrown vertically downward from a height of \(20\) 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, \(g=10~\mathrm{ms^{-2}}\))
1. \(14\) ms^–1
2. \(20\) ms^–1
3. \(28\) ms^–1
4. \(10\) 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 \(2\) kg. Hence the particle is displaced from position \((2 \hat{i} + \hat{k})\) meter to position \((4 \hat{i} + 3 \hat{j} - \hat{k})\) meter. The work done by the force on the particle is:

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