# Energy + momentum conservation paradox

1. Nov 26, 2007

### nocloud

here's the situation

one ball with mass m1 and velocity v1 collides with second ball with mass m2 and 0 velocity. they stick together and the resulting blob has mass m1+m2 and velocity v3 which can be easily found using m1v1+0 = (m1+m2)v3 by conservation of momentum

lets assume we are on level surface and potential energy is zero.

conservation of energy tells us that
.5m1v1^2+0 = .5(m1+m2)v3^2

thus, we have
v1/v3 = (m1+m2)/m1
and
v1/v3 = Sqrt[(m1+m2)/m1]

Can anybody tell me what I am forgetting to consider here?

2. Nov 26, 2007

### D H

Staff Emeritus
This is an inelastic collision, so kinetic energy is not conserved. Momentum is conserved, and you should be able to show that the collision reduces the kinetic energy of the system. Where did the extra energy go? The sticky collision made the balls get hotter than they were. (Energy is still conserved, but you have to take temperature into account.)

Last edited: Nov 26, 2007
3. Nov 26, 2007

### joshd

Is that simply because they coalesce?

We take collisions to be elastic alot in mechanics questions in maths, but I can't remember whether that was only when they don't coalesce...

I guess it can only be elastic if you are using newton's law of impacts, and e=1?

:-\

4. Nov 26, 2007

### D H

Staff Emeritus
The definition of an elastic collision is one in which the kinetic energy remains unchanged. Kinetic energy can be gained (a ball and a contact explosive; a hyperelastic collision) or lost (a ball and a glob of glue; an inelastic collision) as a result of a collision. A collision in which the objects stick together is the extreme example of an inelastic collision, and hence the special name "perfectly inelastic collision".