What is the minimum velocity with which a body of mass \(m\) must enter a vertical loop of radius \(R\) so that it can complete the loop?
1. \(\sqrt{2 g R}\)
2. \(\sqrt{3 g R}\)
3. \(\sqrt{5 g R}\)
4. \(\sqrt{ g R}\)$\mathrm{}$

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:

Force \(F\) on a particle moving in a straight line varies with distance \(d\) as shown in the figure. The work done on the particle during its displacement of \(12\) m is: 

              

A body of mass \(1\) kg is thrown upwards with a velocity \(20\) ms^-1. It momentarily comes to rest after attaining a height of \(18\) m. How much energy is lost due to air friction?
(Take \(g=10\) ms^-2)
1. \(20\) J
2. \(30\) J
3. \(40\) J
4. \(10\) J

An engine pumps water continuously through a hose. Water leaves the hose with a velocity \(v\) and \(m\) is the mass per unit length of the water jet. What is the rate at which kinetic energy is imparted to water?
1. \(\frac{1}{2}mv^3\)
2. \(mv^3\)
3. \(\frac{1}{2}mv^2\)
4. \(\frac{1}{2}m^2v^2\)