A particle of mass \(m\) is driven by a machine that delivers a constant power of \(k\) watts. If the particle starts from rest, the force on the particle at time \(t\) is:
1. \( \sqrt{\frac{m k}{2}} t^{-1 / 2} \)
2. \( \sqrt{m k} t^{-1 / 2} \)
3. \( \sqrt{2 m k} t^{-1 / 2} \)
4. \( \frac{1}{2} \sqrt{m k} t^{-1 / 2}\)

A car of mass m starts from rest and accelerates so that the instantaneous power delivered to the car has a constant magnitude \(P_0\). The instantaneous velocity of this car is proportional to:
1.  \(t^{\frac{1}{2}}\)
2.  \(t^{\frac{-1}{2}}\)
3. \(\frac{t}{\sqrt{m}}\)
4. \(t^2 P_0\)

An engine pumps water through a hose pipe. Water passes through the pipe and leaves it with a velocity of 2 ms^-1. The mass per unit length of water in the pipe is $100$ $kg$ $m^{-1}.$What is the power of the engine?

1. 400 W

2. 200 W

3. 100 W

4. 800 W

A particle of mass \(M\) starting from rest undergoes uniform acceleration. If the speed acquired in time \(T\) is \(V\), the power delivered to the particle is:
1. \(\frac{1}{2}\frac{MV^2}{T^2}\)
2. \(\frac{MV^2}{T^2}\)
3. \(\frac{1}{2}\frac{MV^2}{T}\)
4. \(\frac{MV^2}{T}\)

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. \(\frac{1}{2}m^2v^2\)