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Re: [Phys-l] a conservation equation




On Mar 17, 2010, at 10:25 AM, LaMontagne, Bob wrote:

I was looking at the usual derivation of conservation of energy for a ball thrown in the air. One starts with F=ma, rewrite as F=m dv/dt = m dv/dy dy/dt = m v dv/dy, and then rearrange to Fdy = mv dv. Let the force F be gravity (-mg) and integrate the rearranged version of Newton's Law from y1 to y2 and v1 to v2 to get

mgy1 + KE1 = mgy2 + KE2

One can do a similar derivation using t. Start with F = m dv/dt, rearrange to Fdt=mdv (a vector equation), let the force be -mgj (j is unit vector along vertical) and integrate from t1 to t2 and v1 to v2 (vectors). One obtains

mgt1 j + mv1 = mgt2 j +mv2

or

mgt1 j + p1 = mgt2 j + p2 (vector p's)

This is formally similar to the conservation of energy equation, the difference being that it is a vector equation and that it involves momentum and a term mgt analogous to mgy. The energy equation conserves the sum of KE and a positional energy. The other equation conserves the sum of momentum and a temporal term.

Is anyone aware of an attempt to develop a momentum-time conservation approach to physics similar to the familiar KE-position conservation approach - i.e., an attempt to avoid an impulse-momentum approach by using a conservation law involving momentum and a temporal potential of sorts similar to an avoidance of a work-KE approach by using conservation of KE and positional energies? I would assume that the biggest impediment would be finding suitable forces that are functions of time instead of position.

Bob at PC

Howdy,

This is a Impulse-Momentum equation (well... what you get if the only external force acting on the system is the constant mg in the vertical direction). If the Net External Force on the system is zero in the time interval from t1 to t2 the momentum of the system will be conserved.

Good Luck,

Herb Schulz
(herbs at wideopenwest dot com)