Motion in Two or Three Dimensions

This topic stretches kinematics from one axis to two. A position vector now carries components on both $\hat{\text{ı}}$ and $\hat{\text{ȷ}}$: $\vec{r}(t) = x(t)\,\hat{\text{ı}} + y(t)\,\hat{\text{ȷ}}$. Velocity and acceleration are vector functions of time, found by differentiating $\vec{r}(t)$ component-by-component: $\vec{v}(t) = \dfrac{dx}{dt}\,\hat{\text{ı}} + \dfrac{dy}{dt}\,\hat{\text{ȷ}}$ and $\vec{a}(t) = \dfrac{d^2x}{dt^2}\,\hat{\text{ı}} + \dfrac{d^2y}{dt^2}\,\hat{\text{ȷ}}$. The unit vectors are constants for a fixed Cartesian basis in a nonrotating inertial frame, so they pass through derivatives unchanged. The headline idea is the independence of perpendicular axes: if no force has an $\hat{\text{ı}}$-component, the $\hat{\text{ı}}$-motion is unaffected by anything happening along $\hat{\text{ȷ}}$. The two axes share only the time $t$.

Almost all of it comes from letting the two axes talk to each other when they shouldn't. Three slips in particular: thinking the horizontal velocity drops because the ball is falling, forgetting a component when you differentiate or integrate a vector function, and mixing up speed (a number) with velocity (a vector). The fix is the same in all three: do every operation component-by-component, then combine.