How to find position of a particle at a time given a position dependent force
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.
1 answer
In general, the only way to do it is to solve the equation of motion. In simple cases, that can be done analytically (that is, you can find an explicit formula, but in most cases (outside problems given to students) you have to either make approximations (that is, essentially find a sufficiently close simpler problem that you can solve, and then estimate the error you made due to that simplification), or solve the equation numerically (that is, essentially simulate the system).
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.
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