Phys03

# Phys03 - Unit 3 3.1 3.2 3.3 3.4 Momentum Conservation of...

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1 Unit 3 Momentum 3.1. Conservation of Momentum 3.2 Collisions 3.3 Impulse 3.4 Coefficient of restitution ( e ) 3.1. Conservation of Momentum Consider that we are performing a collision experiment with two particles (not necessary identical particles) on a two-dimensional plane, say, smooth table. If the initial velocity vectors of the two particles were labeled as 1 u G and 2 u G respectively, then after collision, their velocity were found to be 1 v K and 2 v K respectively. The theory behind the collision During the collision, the forces act on each other are with the same magnitude but opposite in direction. This is the Newton’s third law, it is about the action and reaction forces. They are always opposite in directions but they have the same magnitudes (e.g. 12 FF =− G G ). Hence, we have t v m t v m = 2 2 1 1 G G 2 2 1 1 v m v m G G = m 2 m 2 m 1 m 1 1 u G 2 u G 1 v K 2 v K 1 1 1 u v v G K K = 2 2 2 u v v G K K = Before Collision After Collision

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2 Substituting v 1 , v 2 and rearrange the equation, we obtain ) ( ) ( 2 2 2 1 1 1 u v m u v m G G G G = or 11 1 2 2 2 () ( ) 0 mv u +− = G GG G . That is, 2 2 1 1 2 2 1 1 v m v m u m u m G G G G + = + The above expression is the conservation of momentum. Define the momentum of the particle as p G , where p mv = . We can rewrite the above equation as constant = i i p K or in another form 0 = i i p K . Experimental facts After performing numerous trials with different initial velocities and final velocity being measured, it was found that: (1) 1 v K is always in opposite direction of K v 2 (2) 1 2 v v = K K constant We can repeat the experiment by changing different particles and we found that different particles have different degree of resistance to change its magnitude of the velocity after the collision. We can check that the constant is given by the ratio of m 2 and m 1 : 1 2 2 1 m m v v = K K where m 1 and m 2 are then called the inertia mass of the particles, which is a measure of the resistance to change the velocity magnitude during an interaction with another particle. From this experiment, we also discover a conservation law if we define a physical quantity called ‘momentum’ by: v m p K K = .
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Phys03 - Unit 3 3.1 3.2 3.3 3.4 Momentum Conservation of...

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