Find unknown rate or constant

A question is this type if and only if it requires determining an unknown parameter (such as ejection rate c or constant k) by substituting given conditions (e.g. speed at a specific time) into the integrated variable mass equation.

1 questions · Challenging +1.2

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Edexcel M5 2013 June Q3
14 marks Challenging +1.2
  1. A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the spacecraft. The burnt fuel is ejected at a constant rate of \(c \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds the total mass of the spacecraft, including fuel, is \(m \mathrm {~kg}\) and the speed of the spacecraft is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that, while the spacecraft is ejecting burnt fuel,
    $$m \frac { \mathrm {~d} v } { \mathrm {~d} t } = 2000 c$$ At time \(t = 0\), the mass of the spacecraft is \(M _ { 0 } \mathrm {~kg}\) and the speed of the spacecraft is \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 50\), the spacecraft is still ejecting burnt fuel and its speed is \(6000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find \(c\) in terms of \(M _ { 0 }\).