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proton's acceleration

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Using the Coulomb's law, the force between two protons inside a nucleus is about 14.4 N
Isn't that a colossal force for such a tiny particle?. I guess that shows how strong nuclear force must be to overcome this.

When I divide this however to the mass of proton, the acceleration due to Newton's law comes up as

=8.6 x 10^27 m/sec^2

That's far far beyond the speed of light for 1 second of acceleration, even for a fraction of second.

Doesn't that contradict with the special reality that says nothing goes beyond the speed of light??

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