Elastic Collisions and conservation of momentum

[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?

 

[STAii]

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 :
m₁v_1i+m₂v_2i=m₂v_1f+m₂v_2f
0.5m₁v_1i²+0.5m₂v_2i²=0.5m₁v_1f²+0.5m₂v_2f²
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 v_1f and v_2f) 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)

 

[Rockazella]

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?


If this collision weren't elastic, you mentioned it would be impossible without more info to find the final velocity ...what more info?

 

[STAii]

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.

 

[dg]

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