On finding the coefficient of restitution.?

Question

a ball of mass m1 strikes another ball of mass m2 at rest with velocity 'u'. the first ball comes to rest and the second ball moves away with some velocity "v". the kinetic energy lost is half the initial kinetic energy of the first ball (the collission is not elastic)

Answer 1

Answer 1 is mistaken in saying no energy is lost. What is conserved is momentum, but in a not perfectly elastic (c<1) collision, kinetic energy is lost. Also, the coefficient of restitution c is the ratio of velocity differences post- and pre-collision. Bounce height ratio gives the ratio of energies, and must have its square root taken to yield c.
The problem is a little unclear. Does m1 "come to rest" immediately (i.e., stop dead)? Or does it roll and gradually stop due to friction? Assuming m1 stops dead and m2 is moving:
All m1's momentum is transferred to m2, so m2*v=m1*u, and v/u=m1/m2 (eq. 1). But since half the kinetic energy is lost, 0.5*m2*v^2/(0.5*m1*u^2)=0.5 thus (v/u)^2=0.5*m1/m2 (eq. 2). From eqs. 1 and 2 we get v/u = 0.5. The restitution coefficient is the ratio of final to initial velocity differences, or 0.5. Example masses and velocities: m1=1, u=2, ke1=2; m2=2, v=1, ke2=1. The collision is not perfectly elastic, but elasticity is present. The reference has a good iscussion of collision kinetic energy.

Answer 2

I have problems also. What i can't understand is that sometimes when finding the speed of approach/seperation , the two values are added , and in seemingly similar questions they are subtracted. It is somewhat confusing to me as to when we
should we add or subtract the two values?

However, if we study the gravitational aspects.. of a ball falling, etc...one of the things that takes the most kinetic energy out of a ball is impacts. The amount of energy conserved when two surfaces strike each other can be found with the coefficient of restitution between the surfaces. This is one of the more important concepts in our sculpture because one of the main elements is two balls colliding midair. The coefficient of restitution is a ratio of the velocity right after it hits (the surface) divided by the velocity directly after it hits:

c = Coefficient of Restitution

c = v2/v1

An easier way of finding the Coefficient of Restitution, however, is by measuring the first drop height and dividing it by the peak height of the first jump:

c = h2/h1

A low coefficient of restitution constitutes high energy loss, while a high coefficient of restitution means low energy loss. Actually, no energy is being "lost". The energy is simply transferred from the ball to the object you are hitting, giving the other object a tiny push.

Answer 3

The coefficient of restitution (denoted by the symbol c in our formulas) is a measure of the elasticity of the collision between ball and racquet. Elasticity is a measure of how much bounce there is, or in other words, how much of the kinetic energy of the colliding objects before the collision remains as kinetic energy of the objects after the collision. With an inelastic collision, some kinetic energy is transformed into deformation of the material, heat, sound, and other forms of energy, and is therefore unavailable for use in moving.

A perfectly elastic collision has a coefficient of restitution of 1. Example: two diamonds bouncing off each other. A perfectly plastic, or inelastic, collision has c = 0. Example: two lumps of clay that don't bounce at all, but stick together. So the coefficient of restitution will always be between zero and one.

The coefficient of restitution is the ratio of the differences in velocities before and after the collision. In other words, the difference in the velocities of the two colliding objects after the collision, divided by the difference in their velocities before the collision. In symbolic language:

(s2 -v2)=c*(v1-s1)



c = coefficient of restitution

v1 = linear velocity of the racquet mass center before impact

s1 = linear velocity of the ball before impact (will be negative according to our convention that away from the player is positive)

v2 = linear velocity of the racquet mass center after impact

s2 = linear velocity of the ball after impact

From the coefficient of restitution formula, it follows that

c= (s2 - v2)/ (v1 - s1)

To find the coefficient of restitution in the case of a falling object bouncing off the floor, or off a racquet on the floor, use the following formula:
c =sq.root(h/H)


c = coefficient of restitution (dimensionless)

h = bounce height

H = drop height

For the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness which could add or subtract from the coefficient of restitution.