When we expand from one dimension of motion into 2 or more dimensions, we can consider the motion in each dimension separately. Note that we can do this for the standard $x$ and $y$ directions, but also in some other ways, like tilted axes on an inclined plane.

As an example, standard projectile motion problems will often solve for the time of motion via analysis on the $y$ direction, and then using that information to explore the motion in the $x$ direction. Projectiles moving in a uniform gravitational field (as is usual on Earth) will move in a parabola.

1. Constraints and projections: Since a rigid rod cannot compress and an object cannot sink into a plane, you may often need to use common reasoning to impose constraints on objects and their speeds. For example, the velocity component along the rod for the ends of a rigid rod must be the same.
2. Rotated reference frames: You'll often need to use this trick in problems that involve an inclined plane. Rotate the reference frame and split gravity into components parallel ($g\sin\theta$) and perpendicular ($g\cos\theta$) to the plane.
3. Moving reference frames: As described in the last lecture, the motion of multiple objects in free-fall can be simplified by putting the objects in an accelerated reference frame where gravity disappears. In situations involving friction, we learned this week that it's helpful to shift into the rest frame of the medium creating the friction.
4. Reflections: When an object bounces perfectly elastically on a surface (i.e. no loss of energy), you can find an ``image" trajectory to be identical to as if it had passed through the surface unaffected. Then, just reflect the trajectory across the surface to find the actual trajectory.