Elastic collisions and Newton's law of restitution
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Elastic collisions and Newton's law of restitution
It is common sense that objects, and substances, differ in their degree of elasticity. For example, an "elastic" ball will bounce back up when dropped onto the floor, but a piece of putty will stick to the floor instead. Generally, the bouncy ball will not bounce back to the full height from which it was dropped. If it did bounce back to the full height it would be said to be perfectly elastic. For a ball, this would be remarkable - such a ball, once dropped, would carry on bouncing forever. We would not expect this of a ball, because we would expect some of the energy of the ball to be lost with each bounce. However, some collisions do come close to being perfectly elastic. Hard metal spheres when they collide in a pendulum style arrangement will carry on 'clicking' for a long time. [Diagram goes here - download the original to see it.] At the other end of the scale, the piece of putty that 'sticks' to the floor is perfectly inelastic. Suppose we drop an elastic (but not perfectly elastic) ball and video the result; suppose also that the ball is bouncing in a vacuum chamber, so we can ignore the effects of air resistance, then it is law of physics that we expect to observe that the height reached by the ball is a constant proportion of the height from which it fell. For example, suppose the proportion is half. Then we would expect to observe a ball that was also travelling forward with a constant speed, make the following trajectory: [Diagram goes here - download the original to see it.] Newton discovered this trajectory to be an empirical law of nature. It means that the elasticity of an object is a constant property - or rather, elasticity that is a constant property of the contact between surfaces, remains constant over a wide range of conditions. Since the height gained by an object is proportional to its kinetic energy, which is related to its speed, it makes sense to define elasticity in terms of the speed of impact of two objects and the speed of separation. The speed of impact is also called the approach speed. We already use the term 'elasticity' to describe, for example, the response of a spring or elastic band to stretching. Consequently, to distinguish elasticity in that sense from elasticity in the sense being discussed here - the context of collisions - we use the term "coefficient of restitution".
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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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