Linear motion of a body of variable mass
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Linear Motion of a Body of Variable Mass - Rockets and raindrops
Up to this stage we have only considered the laws governing the motion of particles of constant mass. But in many cases the mass of a particle will change as it moves. The two standard examples are rockets and raindrops. [Diagram goes here - download the original to see it.] As the rocket moves forward it expels gas. The loss of gas makes the rocket lighter. [Diagram goes here - download the original to see it.] As the raindrop falls through a vapour cloud more of the vapour adheres to it - it gains mass. The motion of the a missile is an instance of Newton's Third Law - the missile travels forward because vapour is projected from its rear end. The missile and the vapour are recoiling from each other - total momentum is conserved. [Diagram] A missile of mass M moves with speed V, ejecting gas of mass m' at speed v'. Conservation of momentum tells us that: Mv = m'v' Here we are using only magnitudes, but strictly the velocities of the rocket and the expelled gases are vectors and should be measured according to a sign convention - for example, that movement to the right is positive, to the left negative. Hence [Diagram] Then total momentum is conserved [Equation] However, since the mass of the rocket is constantly changing and the gas is ejected continuously, this relationship is not very useful in solving such questions as "how fast is the rocket travelling at a given instance?" At any given instance the rocket exerts a force, F, on the ejected gas. Likewise, in accordance with Newton's Third Law, the gas exerts a force of equal magnitude on the rocket. [Diagram] The force acting on the rocket will cause it to accelerate. The magnitude and direction of this force is given by the "full" version of Newton's Second Law. Up to this time you have probably encountered Newton's 2nd Law in the "restricted" version. That is, as:[Equation] Force = mass ´ acceleration. This version applies only in contexts where the mass of the object is constant. The "full" version is required when we consider linear momentum of a body of variable mass.
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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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