Find ejected or remaining mass

A question is this type if and only if it asks for the mass of fuel ejected or the remaining mass of the system after a given change in speed, rather than asking for a velocity or position.

2 questions · Challenging +1.8

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Edexcel M5 Q5
11 marks Challenging +1.8
5. A spaceship is moving in deep space with no external forces acting on it. Initially it has total mass \(M\) and is moving with speed \(V\). The spaceship reduces its speed to \(\frac { 2 } { 3 } V\) by ejecting fuel from its front end with a speed of \(c\) relative to itself and in the same direction as its own motion. Find the mass of fuel ejected.
(11 marks)
Edexcel M5 2006 June Q5
12 marks Challenging +1.8
A space-ship is moving in a straight line in deep space and needs to reduce its speed from \(U\) to \(V\). This is done by ejecting fuel from the front of the space-ship at a constant speed \(k\) relative to the space-ship. When the speed of the space-ship is \(v\), its mass is \(m\).
  1. Show that, while the space-ship is ejecting fuel, \(\frac{\mathrm{d}m}{\mathrm{d}v} = -\frac{m}{k}\). [6]
The initial mass of the space-ship is \(M\).
  1. Find, in terms of \(U\), \(V\), \(k\) and \(M\), the amount of fuel which needs to be used to reduce the speed of the space-ship from \(U\) to \(V\). [6]