# How to Find Trajectory of Particle?

Let’s say I have a particle, and I know all the forces acting on it at every position. (Let’s say the particle is in an electric/gravitational field to simplify the mathematics involved.) Now, is there a way to find the trajectory of the particle? (i.e. Its position at every instant in time?)

• Do you understand calculus? And what a differential equation is? Commented May 8 at 4:30
• I know all the forces acting on it at every instant in time. Usually you won’t know $\mathbf F(t)$. Instead you’ll know $\mathbf F(\mathbf x, \mathbf v)$. Think about what the laws for gravitational force and electromagnetic force look like. They involve position and velocity, not time. Commented May 8 at 5:04
• This is a huge topic. Sometimes you can find an analytic solution. But generally you need to use a numerical method. See en.wikipedia.org/wiki/… Commented May 8 at 5:05
• $m\ddot{\mathbf x}=\mathbf F(\mathbf x, \dot{\mathbf x})$ is a second-order ordinary differential equation whose solution gives $\mathbf x(t)$. Commented May 8 at 5:10
• In principle you just solve a second order differential equation. In practice this can be difficult to do exactly, or even approximately. Commented May 8 at 6:22

• I didn’t say it was actually wrong; I just thought it was incomplete. (And I still do.) But since you’re new here, and since it does at least cover a case like a uniform time-dependent electric field, where you would know $\mathbf F(t)$, so I’ll reverse my downvote. However, the system won’t let me do that until you make some minor edit. Please notify when when you do. If you decide to explain the differential equation case properly, I’ll upvote. Commented May 9 at 6:31