Since there’s a reasonably strong link to calculus, and mechanics as you’ve already found, it could theoretically help in physics simulations either in a computer or on paper.

As for practical application, well, emulating physics is pretty important in a lot of computer games, or getting robots (assembly line arms, androids, automated vacuum cleaners) around the place and to do what they need without accidentally catapulting themselves into next Tuesday.

How that’s actually programmed might not involve dual numbers at all, but they’re one way of looking at how those calculations might be done.