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Re: momentum before force



Hi Jack Uretsky
You write:

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Hi Gene Mosca and all-
You write:

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I have a problem with the statement F = dP/dt = m dv/dt + v dm/dt. If F,
m, and dv/dt are Galilean invarients and v is not, then this equation
cannot be correct. In fact, the rocket equation -m dv/dt = u dm/dt cannot
be derived from this statement. Note that in the rocket equation m, dv/dt
and u are Galilean invarients. For the original statement F = dP/dt = m
dv/dt + v dm/dt to be correct requires that the mass not be a scalar, as
is the case with the so called relativistic mass m/(1 - (v/c)^2)^1/2.


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Perhaps the following problem will help clarify the issue:
A little red wagon with a mass of 2 kg is filled with 5 kg of
sand. The sand is leaking out of a hole in the wagon at a rate of .q
kg/s. A motor attached to the wagon is pushing the wagon forward with a
thrust of 1 N. Calculate the acceleration of the wagoon as a function of t
before the sand is all gone. Assume that the wagon has an arbitrary
velocity v at t=0. Do the calculation both in the frames where the where
the initial velocity was v and where that velocity was 0.
Regards,
Jack


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Jack,

I did the calculation you suggest using the equation Fext = m dv/dt + v
dm/dt. The resulting expression in both frames is dv/dt = (Fext - v
dm/dt)/m. However, in the frame of the road the initial value of v is
greater than zero while in the frame moving with a constant velocity equal
to the initial velocity of the wagon the initial value of v is zero.
Thus, the presumption that Fext = m dv/dt + v dm/dt leads to a
contradiction. (Acceleration is a Galilean invariant so the initial
acceleration should be the same in both frames.)

I also did the calculation using the rocket equation Fext - u dm/dt = m
dv/dt, where u is the velocity of the sand relative to the wagon. If we
let the direction of motion of the wagon be the direction of increasing x
then the x component of u is zero (assuming the hole is through the bottom
of the wagon and not through either the back or front). Solving for the
acceleration using this formula gives dv/dt = Fext/m, a result which is
the same in both reference frames.

A nice example Jack.

Gene