Earlier I wrote:
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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.
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Then you responded:
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But does your result make sense? Let's modify the problem
a bit. We'll turn of the engine and let the little red wagon be
coasting along a level sidewalk with constant velocity V. Then
at time t=0 the famous massless plug falls out of the bottom of
the wagon and the sand begins to drain out as before. Does the
wagon accelerate? If so, how much? Do you get the same
value in both reference frames?
How does the result change if V = 0 at t=0?
Your latest response leaves me confused. It seems to me that
the answers to your latest questions merely special cases of my
earlier responses, with Fext = 0. Using dv/dt = (Fext - v
dm/dt)/m with Fext = 0 gives dv/dt = (- v dm/dt)/m which for an
initially moving wagon gives different values for the initial
acceleration in the two frames. Using dv/dt = Fext/m (the rocket
equation) with Fext = 0 gives dv/dt = 0 in either frame, a result
consistent with acceleration being a Galilean invariant.
What point are you trying to make? From your previous
response I was felt were kindly supplying a concrete example
supporting my earlier statements that criticized the equation F =
dp/dt = m dv/dt + v dm/dt. Now I'm unsure of your intent.