# Elastic Collisions and conservation of momentum

• Rockazella
In summary, the conversation covers the topic of conservation of momentum in a basic physics class. The application of this concept is demonstrated through the example of an elastic collision between two objects. It is explained that in an elastic collision, both the total momentum and the sum of energies in the system are conserved. The equations used to solve for the final velocities of the objects are also provided. The definition of an elastic collision is given as a collision in which there is no lost energy, and it is noted that in real life, there is no perfectly elastic collision. Additionally, it is mentioned that there are other types of collisions, such as completely anelastic collisions, in which the objects lose all of their relative velocity and stick together.

#### Rockazella

In the rather basic physics class I'm in we just coverd conservation of momentum. We went over the application of this with elastic collisions. I understand that Total momentum before the collision = Total momentum afterwards. What I don't uderstand is how can you tell what the individual momentums will be afterwards.

For example: 2 objects fly directly twords each other. Both objects are 5kg and they each are going at 5m/s. Thus the momentum of one of the objects is 25kgm/s and the other -25kgm/s. Total momentum in the system is 0. After they collide, each could move away from each other at 5m/s or 1000m/s or 5434676m/s, the total momentum in the system would still be 0. Obviously in the real world only one of these would be accurate...so what am I missing, how do you finish the problem?

Well, in elastic collision there are two conditions.
First : The momentum is conserved.
Second : the sum up of energies in the system are conserved.

Now if the collision you are talking about is not elastic, it would be impossible to know the final velocity of any of the balls (whithout more information about the balls), but since the collision is elastic, you can know the speed.
Just solve those two equations :
m1v1i+m2v2i=m2v1f+m2v2f
0.5m1v1i2+0.5m2v2i2=0.5m1v1f2+0.5m2v2f2
Notes:
-Remember that momentum is not scalar, it is a vector.
-"1f" means the final velocity of the first ball, "2i" means the initial velocity of the second ball, and so on ..
-You can multiply the second equation by 2 to eliminate the fraction (0.5)

Now if you solve the two equations above (for the unknown v1f and v2f) you will see that the ballw which inital velocity was 5m/s will have a final velocity of -5m/s, and the ball which's initial velocity was -5m/s will have a final velocity of 5m/s (of course this does not work for all problems this way !).
I found those fast answers using a rule (which comes from the two above equations) that says "in an elastic collision, if the two bodies collide in a head-on collision, and they both have the same mass, they will exchange velocities"

Hope i helped.
(edited for a missing number)

Yup, that helps.

Two more questions:

I should have probably looked this up before, but what is the definition of an elastic collision. I know there are many objects that if they ever collided, wouldn't stick to each other, but also wouldn't really bounce apart (theyd absorb much of the impulse) how do figure the outcome of a collision if these objects are involved?

Well, in the problem you are either told that the collision is elastic, or if you find that the energy before and after the collision are equal, then the collision is elastic.
Mainly, the definition of an elastice collision, is a collision in which there is no lost energy .
In real life (macrosopically), there is no elastic collision, but some collisions are so near to be elastic that we consider they are.
But in microsopic life, we find elastic collision between gas particles.
This is the answer of your first question, now to the second one ...
If the collision was not elastic, the problem must contain one of these extra info (i am not sure if these are all of them, but i can't think of any other now).
1-The lost energy
2-The ratio between the relative speeds before and after the collision (this has a name, but i forgot it)
3-The speed of one of the objects after the collision.

Tell me if you want to know more.

Some typical collision problems besides the elstic ones are the so called completely anelastic collisions.

In this ind of collisions the bodies lose the maximum amount possible of energy. That is they completely lose their velocity relative to the center of mass of the system (velocity of the center of mass must be conserved since this is equivalent to conservation of momentum) and afterwards they proceed 'glued' together with the velocity of their center of mass.

Everything in between an elastic collision and a completely anelastic one is simply called an anelastic collision and it is completely defined by assigning the amount of energy lost in the collision.

Hope this helps, Dario

## 1. How is momentum conserved in an elastic collision?

In an elastic collision, the total momentum of the system before the collision is equal to the total momentum after the collision. This means that the sum of the momenta of all objects involved in the collision remains constant.

## 2. What is an elastic collision?

An elastic collision is a type of collision in which the total kinetic energy of the system is conserved. This means that the objects involved in the collision do not lose any energy to heat, sound, or other forms of energy.

## 3. How does the mass and velocity of objects affect elastic collisions?

The mass and velocity of objects involved in an elastic collision affect the outcome of the collision. The larger the mass and/or velocity of an object, the greater its momentum and the more force it can exert on other objects. In an elastic collision, the total momentum of the system is conserved, so the mass and velocity of each object must be taken into account to determine the final velocities after the collision.

## 4. What is the difference between elastic and inelastic collisions?

In an elastic collision, the total kinetic energy of the system is conserved, while in an inelastic collision, some of the kinetic energy is lost due to the objects sticking together or deforming. Inelastic collisions also do not conserve momentum, as some of the kinetic energy is converted into other forms of energy such as heat or sound.

## 5. Can momentum still be conserved in a collision if the objects are not in a closed system?

Yes, momentum can still be conserved in a collision even if the objects are not in a closed system. This is because the total momentum of the objects involved in the collision remains constant, regardless of whether there are external forces acting on them. However, if external forces are present, the total kinetic energy of the system may not be conserved, making the collision inelastic.