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# How to find position of a particle at a time given a position dependent force

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If we have a force which changes depending on the position of a particle, how can we find the position of the particle at some time $t$?

We can find its velocity if it has travelled a given distance

$$\int^{r_f}_{r_o} F(r)dr = \frac{1}{2} \cdot m_p(u_f^2 - u_o^2)$$

but this equation doesn't involve time and I don't see how we can 'generate' time from a position varying force.

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For a single particle in an external force field with a force that depends solely on position, the equation of motion is basically Newton's third law, $\mathbf F=m\mathbf a$, or written as differential equation $$m\frac{\mathrm d^2\mathbf r}{\mathrm dt^2} = \mathbf F(\mathbf r)$$ How to solve this differential equation (and whether it has a closed solution at all) is of course dependent on the exact form of the force.