Chapter03n

Chapter03n - Chapter 3 Momentum and Collisions 3.1 3.2 3.3...

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Chapter 3 Momentum and Collisions 3.1. Conservation of Momentum 3.2 Collisions 3.3 Impulse 3.4 Coefficient of Restitution ( e ) 3.1. Conservation of Momentum Suppose we perform a collision experiment with two (not necessary identical) particles on a two-dimensional plane, say, a smooth table. We label the initial velocities of the two particles as 1 u and 2 u , and their velocities after collision as 1 v and 2 v respectively. After performing numerous trials with different initial velocities and final velocities being measured, it was found that: (1) 1 v is always in opposite direction of v 2 (2) = 2 1 v v constant We can repeat the experiment for different particles and found that different particles have different degree of resistance to change the magnitude of its velocity after the collision. So we can assign the proportionality constant such that: 1 2 2 1 m m v v = 1 Before Collision After Collision m 2 m 2 m 1 m 1 1 u 2 u 1 v 2 v 1 1 1 u v v - = 2 2 2 u v v - =

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where m 1 and m 2 are called the inertial mass of the particles. Note that the inertial mass of a particle is a measure of the resistance to change the magnitude of the velocity during an interaction with another particle. In fact, we can rewrite the above experimental results as: 0 2 2 1 1 1 2 2 1 = + - = v m v m m m v v From this experiment, we also discover a conservation law called the conservation of linear momentum if we define a physical quantity called ‘momentum’ by: v m p = . The mathematics behind During the collision, the forces acting on each of the colliding bodies are with the same magnitude but opposite in direction. This is the Newton’s third law. Hence, we have t v m t v m - = 2 2 1 1 2 2 1 1 v m v m - = Substituting v 1, v 2 and rearranging the equation, we obtain ) ( ) ( 2 2 2 1 1 1 u v m u v m - - = - 0 = i i p That is, 2 2 1 1 2 2 1 1 v m v m u m u m + = + constant = i i p Example 3-1 An ant lands on one end of a floating 4.75 g stick. After sitting at rest for a moment, it runs toward the other end with a speed of 3.8 cm/s relative to the still water. The stick moves in the opposite direction at 0.12 cm/s relative to water. What is the mass of the ant? Answer : The total momentum of the ant-stick system before the ant runs on the stick is zero. By the conservation of linear momentum, the total momentum of the system after the ant runs on the stick equals zero. Hence we can write p a + p s = 0 m a v a + m s v s = 0, Substituting v a, v s and m a , we obtain m a (3.80) + 4.75( - 0.12) = 0 m a = 0.15g 2 Still Water
3.2 Collisions Consider an isolated system made up of two or more objects. (1) In an elastic collision, both the total momentum and total kinetic energy are conserved.

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Chapter03n - Chapter 3 Momentum and Collisions 3.1 3.2 3.3...

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