For a practical application of this form of force rather than the simplistic version, consider the following question.
4.2.1 Consider a rocket that has mass \(m\) and speed \(v\). In the middle of its journey, the rocket starts to eject fuel to fly faster. Let the fuel be ejected at a velocity \(u\) relative to the rocket at a rate of \(\frac{dm}{dt}\). If the rocket is made out of 90% fuel, by what factor will the velocity of the rocket increase after all the fuel is dumped?
4.3 Changing Frames
Note that as long as it is an inertial frame, the conservation of momentum must hold provided there are no external forces. A frame that is often convenient is the centre of mass frame. Consider the classic exploratory momentum question:
4.3.1 A mass \(m\) with speed \(v\) approaches a stationary mass \(M\). The masses bounce off each other without any loss in total energy. What are the final velocities of the particles? Assume all motion take place in 1-D. Use the centre of mass frame. Note that after the collision, both masses must "bounce" with the reverse of their initial velocities.