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I don't know if this type of question has been answered and I don't know how to search for it so I am asking it here. Say we have two positive particles in space, with no other forces acting on each other except for the electromagnetic force. I am trying to find the position of the particles based on time.

The force between these two would be: $$F = \frac{kqq }{r^2}$$ For simplicity, let's hold one of these particles in space, and assume $mkqq = 1$.

Therefore, for the particle not held in place: $$a = \frac{1}{r^2}$$ The problem here is that I cannot substitute my $x = \frac{1}{2}at^2$ formula that I learned in algebra based physics. Since the acceleration isn't constant. If I were to use jerk ($x = \frac{1}{6}jt^3$), it still wouldn't work since I assume it isn't constant either.

I tried to integrate, but I don't have enough calculus knowledge to integrate r here based on dt. I get: $$\iint a(t)dtdt = \iint \frac{1}{r^2}dtdt $$ $$ x(t) = \iint \frac{1}{r^2}dtdt$$ $r$ is equal to $x$ or $x + x_0$

I don't know how to get time into this equation, or how to evaluate the $\iint \frac{1}{r^2}dtdt$ being respect to dt.

Unknown
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1 Answers1

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Hint: if $a =\ddot r = \frac {1}{r^2}$ then $\dot r \ddot r = \frac {\dot r }{r^2}$, therefore $ \frac{1}{2}\frac{d}{dt} (\dot r)^2 = \frac {\dot r }{r^2}$. Can you continue from here?

hyportnex
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