The basic form of velocity = start_velocity + (acceleration) * (time) is valid in any number of spatial dimensions. The only thing that changes is the number of components in the relevant vectors.

All physics stuff I can think of that works on a vector works regardless of how many spatial dimensions you have, 1D, 2D or 3D doesn't really matter.

As ApochPiQ said, it just changes the vector from being e.g. direction.x to direction.x, direction.y, direction.z.

Cross product is 3D specific, that's maths not physics though. There is an analogy for any dimension, the wedge product.

Physics generalizes pretty well to higher dimensions, so the vector functions like v = u + at, F = ma, and so on apply in any number of dimensions. However some things are different due to the changes in geometry. For instance, there is no rotation in the usual sense in 1D, the cross product operation is not closed in neither one nor two dimensions (but you can still define something which works similarly from a physics perspective). While physics in general is typically first defined in three dimensions because, as far as we can tell, this is the space in which we exist, along with a temporal dimension, there is no reason we can't come up with generalizations in lower or higher dimensions where all or most of the known laws of physics still apply in some sense, even if we can't visualize or test them directly. In fact, generalizations often guide physicists to find simplifications or improvements to existing theories, so it's actually beneficial to seek them.

So, to answer your question, yes, people have tried very hard to make sure that all of the physics equations are applicable in as many different situations as possible, and that includes different number of dimensions. However, you may need to make some changes to some of the code between 2D and 3D to account for fundamental changes in how the various operations which physics builds on are defined (and whether they apply at all in a meaningful way, e.g. rotation in one-dimensional space). Overall you can probably think of 2D as being 3D constrained to two dimensions.

However v = u + at probably doesn't "apply" that well in any dimension to find the velocity of a car, since a car generally is not accelerating in a constant direction at a constant rate. You probably want dv = a * dt where the acceleration of the car is integrated over time to obtain its change in velocity (and you can then integrate that to obtain its change in position).

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More formally, modern physics does not assume that space is three-dimensional, or even euclidean. The formulas apply to any type of space-time geometry which meets specific mathematical requirements (which depend on the theory being considered, e.g. classical newtonian physics, general relativity, etc.). If you've learnt polymorphism and/or interfaces and/or contracts, this is the same idea: the Physics class uses a Space interface, such as euclidean R^3, which defines what distances, points, and vectors are, as well as a Time interface which describes the flow of time, and applies the (very generic) laws of physics to these. How these interfaces are implemented doesn't matter as long as they behave as expected by the Physics class. Thus you can create different "kinds" of physics which look different, but which ultimately all behave the same on some level. Apologies if I've used poor terminology, I'm not a physicist, just trying to convey that physics can be generalized. (and, no, those aren't *real* classes to be coded up, they are an analogy that I thought might be easier to understand to someone with a programming background)

Euclidean distance -- Add the squared difference in each dimension. Square root of the sum.

Doesn't matter how many dimensions.

Distance depends on the topology though. The distance between 2 points on the surface of the Earth is not the Euclidean distance unless you build a tunnel. Same kind of thing applies in Manhattan.