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Q&A Find position of a particle at a time

posted 1y ago by celtschk‭  ·  edited 1y ago by celtschk‭

Answer
#2: Post edited by user avatar celtschk‭ · 2021-08-14T13:38:18Z (about 1 year ago)
  • 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 (that is, 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.
  • 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.
#1: Initial revision by user avatar celtschk‭ · 2021-08-14T13:36:43Z (about 1 year ago)
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 (that is, 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.