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Impulse and Collisions Ch 9 Male rams butt heads at high speeds in a ritual to assert their dominance. How can the force of this collision be minimized so as to avoid damage to their brains? Linear Momentum Momentum is a vector; there is a direction and magnitude. Momentum = mass x velocity p = m v In general, we must worry about the x, y, and z components of momentum If the particle is moving with in an arbitrary direction then p must have three components which are equivalent to the component equations: p x = mv x p y = mv y p z = mv z Newton called m v the quality of motion . v v v A block and a ball of the same mass and downward speed v hit the floor. The block falls and sticks to the floor. x y The ball rebounds with the same velocity it hit the floor with. Change in Momentum v x y Change in Momentum We only have to consider the momentum in the y direction. Block p y = p y,f p y,i = 0 m(-v) = mv The change in momentum is mv upward. Ball p y = p y,f p y,I = mv m(-v) = 2mv The change in momentum is 2mv upward. Be careful about the vector nature of momentum!!! Linear Momentum Newton's Second Law, which we have written as F = m a can also be written in terms of momentum, F = m a = m (d v /dt) = d(m v )/dt F = d p /dt The force acting on an object equals the time rate of change of the momentum of that object. Conservation of Momentum Consider two objects that "interact" or collide or have some effect on each other (billiard balls). From Newton's Third Law of motion, we know F 12 = - F 21 Where F 12 is the force exerted by particle 1 on particle 2 and F 21 is the force exerted by particle 2 on particle 1. In terms of momentum, this means d p 1 /dt = - d p 2 /dt d p 1 /dt + d p 2 /dt = 0 d ( p 1 + p 2 ) /dt = 0 d p Tot /dt = 0; p Tot = p 1 + p 2 Conservation of Momentum Lets look at that again! d p Tot /dt = 0 where p Tot = p = p 1 + p 2 = const This statement may also be written as p 1i + p 2i = p 1f + p 2f Since the time derivative of the momentum is zero, the total momentum of the system remains constant . This is Conservation of Momentum Conservation of Momentum is essentially a restatement of Newton's Third Law of Motion. Conservation of Momentum We have said that if the particle is moving with in an arbitrary direction then p must have three components which are equivalent to the component equations. This allows us to rewrite the conservation of momentum as = = = system zf system zi system yf system yi system xf system xi p p p p p p Conservation of momentum: Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant. 1-D inelastic collisions Perfectly inelastic collisions are those in which the two objects stick together. ...
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