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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
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