Elastic collisions and Newton's law of restitution
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The coefficient of restitution
The coefficient of restitution, e, is defined as [Equation goes here - download the original to see it.] As already remarked, it is an empirical law that this ratio remains constant for a wide range of situations. This also makes it possible to measure the coefficient of restitution for the collision between objects of given materials in one situation and generalise to other situations. Questions are set where the coefficient of restitution is known and some aspect of the subsequent trajectory of the particles must be found. Example 1. A ball is dropped on to the ground from a height of 1.5m. The coefficient of restitution for the collision is 0.5. Find the height of the first bounce. For the speed of impact we use the fact that the gravitational potential energy of the ball is converted to kinetic energy. [Equation goes here - download the original to see it.] Then Newton's law of restitution, with e = 0.5 , gives the speed of separation, v, as : [Equation goes here - download the original to see it.] To find the height: [Equation goes here - download the original to see it.] [EXAMPLE goes here - download the original to see it.] [Equation goes here - download the original to see it.] Likewise, the separation speed, v, is found by equating the gain of gravitational energy with the loss of kinetic energy on separation from the surface. [Equation goes here - download the original to see it.] Clearly, this coefficient of restitution remains constant since in establishing it we used only the supposition that the height from which the ball was falling was h, which would apply to any height whatsoever. Having verified the existence of the coefficient of restitution we can go on to look at more complicated problems on collisions. Proposition In any collision of two smooth objects, the impulse of each object on the other is communicated along the line joining the objects' centres of mass. The term "smooth" is introduced in order to eliminate complications introduced by rough surfaces that catch on to each other. If one surface acts as a "hook" to the other surface's "handle" then they will lock on to one another regardless of whether they are elastic or not. [Diagram goes here - download the original to see it.] So these laws only apply to objects with "smooth" surfaces.
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Contents of
Elastic collisions and Newton's law of restitution
1 Elastic collisions. Impulse
2 Elastic collisions and Newton's law of restitution
3 The coefficient of restitution
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