Linear motion of a body of variable mass
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Linear Motion of a Body of Variable Mass. First Case (mass increment)
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An object of mass M is travelling with velocity V and is subject to an external force F. In time ¶t it is joined by a mass ¶m travelling with velocity v'. [Diagram] Let m = m(t) be the mass at time t. Let v = v(t) be the velocity at time t. Let dm be the mass increment after time dt. Let v' be the velocity of the mass increment. Then m + dm is the mass of the coalesced particle at t + dt. Also v + dv is the velocity of the coalesced particle at t + dt. Then: increase in momentum = momentum after - momentum before [Equation] Therefore the rate of change of momentum is approximately given by:- [Equation] Newton's Second Law gives: [Equation] [Equation] Hence the form of the differential equation governing linear motion of a body undergoing an increment of mass is: [Equation] The expression v - v' represents the velocity with which the object is gaining on the mass increment. Setting: [Equation] We can write: [Equation] where u is the speed of the object relative to the mass increment - or, which is now the same thing, the speed of the mass increment relative to the mass. If you could imagine yourself traveling with the object, the mass increment would appear to be traveling with speed u towards you.
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Contents of
Linear motion of a body of variable mass
1 Linear Motion of a Body of Variable Mass - Rockets and raindrops
2 Linear Motion of a Body of Variable Mass. Newton's Second Law
3 Linear Motion of a Body of Variable Mass. First Case (mass increment)
4 Linear Motion of a Body of Variable Mass. Second Case (mass decrement)
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