How is momentum conserved if you lose kinetic energy?

[CWatters]

Perhaps the OP should also read up about system boundaries. These are imaginary lines drawn around part or all of the things you are interested in analysing. Not really a physical line that has location or dimensions, its more a conceptual idea of what you are considering to be "part of your system". Generally everything inside your system boundary will feature in your conservation equations but there might be terms for energy or momentum that leave your system.

Momentum is only conserved by a "closed system" (no forces crossing the system boundary).
Energy is only conserved by a "closed system" (no energy crosses the system boundary).

Typically when someone askes "where does the energy go" or "doesn't this break conservation of momentum" it means they have forgotten about something that either should have been considered as part of the system (eg they drew their system boundary in the wrong place) or it crosses their system boundary (it's not "closed").

Common things people forget about are:

Energy that escapes the system as heat, perhaps due to friction, deformation of materials or an unaccounted for chemical reaction.
Momentum that escapes the system by transfer to/from planet Earth.

So when considering two objects that collide, if you want the make statements like "momentum is/isn't conserved" you first need to say what you are including in your system boundary... Are you talking about the momentum of one of the objects or both of the objects? Is it an elastic collision (no energy lost as heat) or inelastic (energy lost as heat).

PS: There are probably better more formal ways to describe system boundaries).

 

[CWatters]

Specifically it is my understanding that every time you start or stop moving (or accelerate) on Earth, you change the Earth's rotation rate.

Back of the envelope calculation...

10,000kg truck accelerates from rest to 50mph (22m/s) on the Earth which has a radius of 6,371,000 meters

Angular momentum L = Iω or rmv
where
r is the radius of the earth
m mass of truck
v velocity of truck

So the angular momentum of the truck increases to:

L_t = 6,371,000 * 10,000 * 22
= 1.4 * 10¹² kg.m².s⁻¹

Depending on which direction the truck goes this will increase of decrease the angular momentum of the planet by the same amount. Let's assume east or west..

Google suggests the moment of inertia of the planet is around 8 * 10³⁷ kg.m². It rotates at one revolution a day which is an angular velocity of 7.3 * 10^-5 R.s^-1. So the angular momentum of the Earth is about

L_e = 8 * 10³⁷ * 7.3 * 10^-5
= 5.8 * 10³³ kg.m².s⁻¹

So the truck accelerating causes the length of a day to change by about..

= 1.4 * 10¹² * 100 / 5.8 * 10³³
= 2.4*10^-20 %

If it kept going all day without stopping that would make his day longer or shorter by about 2*10^-17 seconds.

PS: I haven't checked my figures very carefully.

 

[robphy]

Possibly interesting reading:
from Edward Purcell's "Back of the Envelope" Problems in AJP
"If every car and truck in the US were driven 300 miles north and left there, by how much would the length of the day be changed?"
http://web.mit.edu/rhprice/www/Readers/Purcell/May1984-Problem1.pdf

...but, of course, this is a digression from the original question.

 

[russ_watters]

Possibly interesting reading:
from Edward Purcell's "Back of the Envelope" Problems in AJP
"If every car and truck in the US were driven 300 miles north and left there, by how much would the length of the day be changed?"
http://web.mit.edu/rhprice/www/Readers/Purcell/May1984-Problem1.pdf

...but, of course, this is a digression from the original question.

Not much of a digression. I considered using the figure skater example earlier, but we weren't dealing with rotation. The figure skater adds or subtracts kinetic energy by extending or retracting her arms (with w=fd against centrifugal force), changing the moment of inertia and rotation rate while keeping angular momentum constant.

Yours is another version of the same problem.

 

[Dr Dr news]

One of my favorite demonstrations in Mechanics is to take two carts of identical mass on a track with velcro affixed to their front bumpers and give them equal and opposite velocities. The system of two carts thus have zero momentum before colliding and stopping in the middle of the track. The momentum is conserved but all of the kinetic energy is converted to thermal energy. Another favorite is to take a mass of lead shot in a cardboard tube and repeatedly invert the tube doing work on the shot, converting the work to potential energy to kinetic energy ending in an inelastic collision, pour the shot into a calorimeter and measure the increased temperature. N(mgh) = mcΔT. The work is ultimately converted to an increase in the internal energy of the shot.

 

[rude man]

I vote for post 2. You will never understand until you understand post 2.
2 masses collide: one might go backwards!

 

[A.T.]

I vote for post 2. You will never understand until you understand post 2.
2 masses collide: one might go backwards!

Even if nothing goes backwards, an inelastic collision still conserves only momentum, not kinetic energy. See example in port #5.

 

[gmax137]

Even if nothing goes backwards, an inelastic collision still conserves only momentum, not kinetic energy. See example in port #5.

Yes, but the gist of the OP seemed to be, how can energy (a function of v) change, without momentum changing (since it is also a function of v). At least that's how i read it.

 

[rude man]

Even if nothing goes backwards, an inelastic collision still conserves only momentum, not kinetic energy. See example in port #5.

Make that "... even go backwards!"