Re: Momentum conservation

On Aug 18, 7:20 pm, "Timo A. Nieminen" <t...@xxxxxxxxxxxxxxxxx> wrote:
On Sat, 18 Aug 2007, Peter wrote:
Hi! When a point object, like a steel ball, collides (without rolling)
in one dimension with another identical object at rest, and stops on
impact, the target object is supposed to acquire the momentum of the
incident object. However, in all real collisions some heat and noise
is always generated, which, of course, is energy that is dissipated.
Where does this energy come from if momentum is conserved?

From the kinetic energy. This means that object 1 _doesn't_ stop due to
the collision alone. In an almost elastic collision (eg, balls in a
Newton's cradle, billiard balls, etc), only a very small amount of energy
is lost, and ball 1 moves very slowly after the collision. With some
friction, it can stop very shortly after the collision.

Consider the limit, the completely inelastic collision. Both objects end
up stuck together, moving along at the same speed (half of the original
speed, since the objects are identical). Clearly, object 1 doesn't stop.
Momentum is still conserved.

Most collisions will be somewhere between the two cases.

How could
momentum be conserved, if this energy comes at the expense of the
kinetic energy of the object?

Momentum is conserved, kinetic energy is, in general, not conserved. Why?

Newton 2 and Newton 3 tell us that when two objects interact, they exert
forces equal in magnitude and opposite in direction for the same time, and
therefore their changes in momentum are equal in magnitude, and opposite
in direction. Newton 1 tells us that momentum is conserved when there are
no interactions. Conservation of momentum is the essence of Newton's laws
of motion.

The work done on an object moving a distance x in a straight line by a
constant force F is F.x (that's the scalar product, or dot product). If
the force and motion are parallel, the work is either Fx or -Fx. Consider
a collision between two balls, with loss of energy. When the balls
interact, the time that ball 1 exerts a force on ball 2 for must be the
same as the time that ball 2 exterts a force on ball 1 for - thus,
momentum is conserved. If the distance each ball moves during the
interaction is identical, and the force is constant during the collision,
then no energy can be lost because work1 = - work2.

Think of the two balls as lossy springs. If an ideal ball collides with a
fixed ideal spring, the spring will exert a force on the ball as it
compresses the spring, and then exert an identical force on the ball as it
throws it back out - the work done by the stopping force as the spring is
compressed and the returning force are the same. If energy is lost, it
must take more force to compress the spring than the spring returns when
it extends again. This will happen with the two balls is a lossy
collision - the force as the two balls collide and compress each other is
greater than the force when they spring apart. The first ball moves
more during the compressing part of the collision, the second ball moves
more during the restoring part of the collision - the energy lost by the
first ball is more than the work done on the 2nd ball.

--
Timo Nieminen - Home page:http://www.physics.uq.edu.au/people/nieminen/
E-prints:http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits:http://www.users.bigpond.com/timo_nieminen/spirits.html

Thanks. I think you are completely right. Would you know if there
exists experimental verification that corroborate what you say? I
realize the amounts to be measured are extremely small.

Peter

.

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