Linear motion of a body of variable mass
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Linear Motion of a Body of Variable Mass. Second Case (mass decrement)
An object of mass m is traveling with velocity v and is subject to an external force F. In time dt it loses a mass dm traveling with velocity v'. [Diagram] [Equation] Then: increase in momentum = momentum after - momentum before. [Equation] Hence: [Equation] Hence, applying Newton's 2nd Law: [Equation] Let us compare the governing differential equations in the two cases: Mass increment: [Equation] Mass decrement: [Equation] We can see that the two terms differ only in the expression v - v' (mass increment) or v' - v (mass decrement). Note v' - v = -(v - v'). v - v' is the velocity of the object relative to the particle. Mass increment: [Diagram] The object appears to be gaining on the mass increment. Mass decrement: [Diagram] The object appears to be moving away from the mass decrement. We set u = | v - v' | and write the equation for both cases - increment and decrement - as:- [Equation] Then u is the speed of the mass increment or decrement relative to the object. It is no longer a vector and it is always understood to be a positive quantity. It is understood to be the speed with which coalescing particles travel towards the object in the case of mass increment, or the speed with which ejected particles travel away from an object in the case of mass decrement. A further note. The term F refers to external forces acting on the system. If, for example, a rocket is running on a frictionless track, then there are no external forces and F = 0. This is the case in our first example. We now proceed to illustrate the application of this equation to specific examples. Example (2) A missile, initially of total mass M0 kg, is launched horizontally from rest along a smooth, frictionless track. [Diagram] Gas is ejected from the missile at a rate of k kg per second and with a speed u ms-1 relative to the missile. If the mass of the missile is m at a time t and its velocity is v at time t, find v in terms of M0, k, u and t. Given u = 20ms-1 the particle ceases to eject gas when its mass is ½M0. What is its maximum speed? The mass ejected in t seconds = kt. \ M = M0 - kt. Newton's Second Law gives: [Equation] Here, there are no external forces, since the track is smooth, hence F = 0; substituting m = M0 - kt [Equation] When t = 0, v = 0, hence c = u ln M0 [Equation] [Diagram] You may be asked to derive the equation of motion from first principals. Another example further illustrates this. Example (3) A rocket ejects fuel at a rate l. The fuel is ejected backwards from the rocket with a constant velocity V relative to the rocket. Use conservation of linear momentum to show that [Equation] The fuel is being used at the rate [Equation] To solve this problem picture the rocket after it has ejected fuel, dm, at a speed V relative to the rocket. [Diagram]increase in momentum backwards = increase in momentum forwards [Equation] We now illustrate the case where external forces apply. In this third example a rocket is moving under gravity. Example (4) A rocket with initial mass M0 is launched from rest. Burnt fuel is ejected downwards at a constant rate M0k kg s-1. The velocity of the fuel relative to the rocket is u ms-1. Ignoring air resistance (i) show that [Equation] and (ii) demonstrate that the rocket cannot be launched unless ku > g. (iii) find v in terms of t. [Diagram] At time t = M(t) = M0 v(t) = 0 v The rocket is moving under gravity. Since we are ignoring air-resistance the weight of the rocket is the only external force acting. The mass of the rocket at time t is given by [Equation] Newton's Second Law states. Equation] [Equation]Shown. (ii) To start moving [Equation Shown. (iii) We have v = 0 when t = 0. The equation is: [Equation] Substituting v = 0, t = 0,we obtain c = 0. [Equation]
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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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