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In introductory mechanics, the momentum of a particle is its mass times its velocity. In electrodynamics, the momentum of a field is proportional to the cross-product of the electric field with the magnetic field. In special relativity, momentum is generalized to four-momentum.

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Is the average force calculated from $F(x)$ the same as that calculated from $F(t)$?

Your reasoning is valid with the important caveat that you yourself realized: this works if $x_1$ corresponds to $t_1$ and $x_2$ corresponds to $t_2$, and as long as the force over time and the force …
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