The motion of an object acted on by a time-dependent, position-dependent force?

[breakfight]

Is there a non-iterative way to solve for the motion of an object under the influence of a time-dependent, position-dependent force?

Here is an example problem:

A bead on a straight wire is subjected to a net force.
There is no friction between the bead and the wire.
The bead and the wire do not experience any gravitational forces.
The bead has a mass of 1kg.

The net force acts along the wire, and is position and time dependent. It can be represented as

F = 9x² + 5xt - 10t²

The bead starts at rest at x = 0.

Where is the bead after 30 seconds have elapsed?​

How would one go about solving a problem like this numerically?

 

[danielakkerma]

Hi,
Sadly there's no way around it :(...(due to the presence of squared and multiplied elements on the right hand side).
This question however, is very reasonably stable using a simple, forward-Euler method. In other words,
$$\Large<br /> x'(t) = \frac{(x_{n+1}-x_n)}{\delta t}<br /> x''(t) = \frac{x'_{n+1}-x'_{n}}{\delta t}$$
And substituting accordingly leads to very accurate and tolerable results.
Have you tried using Mathematica, or Matlab? This can also be effectively modeled in C/C++ or any other programming language.
I hope that helps,
Daniel

 

[Ray Vickson]

Sadly, the forward Euler method is often unstable and should almost never be used for solving a DE over s long interval. That's why so many alternative methods (such as Runge-Kutta, etc.) have been developed.

RGV

 

[danielakkerma]

Oh, I definitely agree... but for short intervals, as is required in the query, as well as just for introductory means, it should work well enough...
Of course, moving on to Runge-Kutta, or some implicit methods should elucidate any issues with accuracy, stability, so forth...
Daniel