- What is the coefficient of restitution for?
- Momentum
- Energy and coefficient of restitution
- How is the coefficient of restitution calculated?
- Example
- Solution
- References
The coefficient of restitution is the quotient between the relative velocity of retreat and the relative velocity of approach of two colliding bodies. When the bodies are united after the collision, this quotient is zero. And unity is worth in the case that the collision is perfectly elastic.
Suppose two solid spheres of mass M1 and mass M2 respectively that collide. Just before the collision the spheres had velocities V1 and V2 with respect to a certain inertial frame of reference. Right after the collision their speeds change to V1 ' and V2'.
Figure 1. Collision of two spheres of masses M1 and M2 and their coefficient of restitution e. Prepared by Ricardo Pérez.
Bold type has been placed in the speeds to indicate that they are vector quantities.
Experiments indicate that every collision fulfills the following relationship:
V1 ' - V2' = -e (V1 - V2)
Where e is a real number between 0 and 1, called the coefficient of restitution of the collision. The above expression is interpreted like this:
The relative velocity of two particles before the collision is proportional to the relative velocity of the two particles after the collision, the constant of proportionality is (-e), where e is the coefficient of restitution of the collision.
What is the coefficient of restitution for?
The usefulness of this coefficient lies in knowing the degree of inelasticity of a collision. In the case that the collision is perfectly elastic the coefficient will be 1, while in a completely inelastic collision the coefficient will be equal to 0, since in this case, the relative speed after the collision is zero.
Conversely, if the coefficient of restitution of a collision and the velocities of the particles before it are known, then the velocities after that collision occurs can be predicted.
Momentum
In collisions, in addition to the relationship established by the coefficient of restitution, there is another fundamental relationship, which is the conservation of momentum.
The momentum p of a particle, or momentum as it is also called, is the product of the particle's mass M and its velocity V. That is, the momentum p is a vector quantity.
In collisions the linear momentum P of the system is the same just before and just after the collision, because the external forces are negligible compared to the brief but intense internal interaction forces during the collision. But the conservation of the momentum P of the system is not enough to solve the general problem of the collision.
In the previously mentioned case, that of the two colliding spheres of masses M1 and M2, the conservation of linear momentum is written like this:
M1 V1 + M2 V2 = M1 V1 ' + M2 V2'.
There is no way to solve the collision problem if the coefficient of restitution is not known. Conservation of momentum, while necessary, is insufficient to predict speeds after collision.
When a problem states that the bodies remain moving together after the collision, it implicitly says that the coefficient of restitution is 0.
Figure 2. In billiard balls there are collisions with a coefficient of restitution little less than 1. Source: Pixabay.
Energy and coefficient of restitution
The other important physical quantity involved in collisions is energy. During collisions there are exchanges of kinetic energy, potential energy, and other types of energy, such as heat energy.
Before and after the collision, the potential energy of interaction is practically zero, so the energy balance involves the kinetic energy of the particles before and after and a quantity Q called dissipated energy.
For the two colliding mass spheres M1 and M2, the energy balance before and after the collision is written as follows:
½ M1 V1 ^ 2 + ½ M2 V2 ^ 2 = ½ M1 V1 ' ^ 2 + ½ M2 V2' ^ 2 + Q
When the interaction forces during the collision are purely conservative, it happens that the total kinetic energy of the colliding particles is conserved, that is, it is the same before and after the collision (Q = 0). When this happens the collision is said to be perfectly elastic.
In cases of elastic collisions, no energy is dissipated. And also the coefficient of restitution fulfills: e = 1.
On the contrary, in the inelastic collisions Q ≠ 0 and 0 ≤ e <1. We know, for example, that the collision of billiard balls is not perfectly elastic because the sound that is emitted during the impact is part of the dissipated energy.
For a collision problem to be perfectly determined, it is necessary to know the coefficient of restitution, or alternatively the amount of energy dissipated during the collision.
The coefficient of restitution depends on the nature and type of interaction between the two bodies during the collision.
For its part, the relative velocity of the bodies before the collision will define the intensity of the interaction and hence its influence on the coefficient of restitution.
How is the coefficient of restitution calculated?
To illustrate how the coefficient of restitution of a collision is calculated, we will take a simple case:
Suppose the collision of two spheres of masses M1 = 1 kg and M2 = 2 kg moving on a straight rail without friction (as in Figure 1).
The first sphere impinges with initial velocity V1 = 1 m / s on the second which is originally at rest, that is, V2 = 0 m / s.
After the collision they are moving like this: the first stops (V1 '= 0 m / s) and the second moves to the right with speed V2' = 1/2 m / s.
To calculate the coefficient of restitution in this collision we apply the relation:
V1 '- V2' = -e ( V1 - V2 )
0 m / s - 1/2 m / s = - e (1 m / s - 0 m / s) => - 1/2 = - e => e = 1/2.
Example
In the one-dimensional collision of the two spheres of the previous section, its coefficient of restitution was calculated, resulting in e = ½.
Since e ≠ 1 the collision is not elastic, that is, the kinetic energy of the system is not conserved and there is a certain amount of dissipated energy Q (for example, heating of the spheres due to the collision).
Determine the value of the energy dissipated in Joules. Also calculate the percentage fraction of energy dissipated.
Solution
The initial kinetic energy of sphere 1 is:
K1i = ½ M1 V1 ^ 2 = ½ 1 kg (1 m / s) ^ 2 = ½ J
while that of sphere 2 is zero because it is initially at rest.
Then the initial kinetic energy of the system is Ki = ½ J.
After the collision, only the second sphere moves with velocity V2 '= ½ m / s, so the final kinetic energy of the system will be:
Kf = ½ M2 V2 '^ 2 = ½ 2 kg (½ m / s) ^ 2 = ¼ J
That is, the energy dissipated in the collision is:
Q = Ki - Kf = (½ J - ¼ J) = 1/4 J
And the fraction of energy dissipated in this collision is calculated as follows:
f = Q / Ki = ¼ / ½ = 0.5, that is to say that 50% of the energy of the system has been dissipated due to the inelastic collision whose coefficient of restitution is 0.5.
References
- Bauer, W. 2011. Physics for Engineering and Sciences. Volume 1. Mc Graw Hill.
- Figueroa, D. 2005. Series: Physics for Sciences and Engineering. Volume 1. Kinematics. Edited by Douglas Figueroa (USB).
- Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
- Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1.
- Wikipedia. Amount of movement Recovered from: en.wikipedia.org.