Motion in Two and Three Dimensions

In one dimension a particle's location is given by a single coordinate. In two or three dimensions we use a position vector $\vec{r}$ that extends from the coordinate origin to the particle:

The coefficients $x$, $y$, $z$ are the rectangular coordinates of the particle — they are also the scalar components of $\vec{r}$. As the particle moves, $\vec{r}$ continuously points from origin to particle, sweeping through space.

When a particle moves from position $\vec{r}_1$ to $\vec{r}_2$, its displacement is the vector change:

If a particle undergoes displacement $\Delta\vec{r}$ in time $\Delta t$, its average velocity is the displacement divided by elapsed time:

As $\Delta t \to 0$, this limit is the instantaneous velocity — the time derivative of the position vector:

The speed is the magnitude of $\vec{v}$: $v = \sqrt{v_x^2 + v_y^2 + v_z^2}$. To find $\vec{v}$ from a position function $\vec{r}(t)$, differentiate each scalar component separately with respect to time.

Whenever the velocity vector $\vec{v}$ changes — whether in magnitude, direction, or both — there is an acceleration. The average acceleration over time interval $\Delta t$ is:

The instantaneous acceleration is the derivative of $\vec{v}$, equivalently the second derivative of $\vec{r}$:

Because acceleration is computed by differentiating each scalar component independently, we can analyze $a_x$, $a_y$, and $a_z$ as separate 1D problems. This is exactly the independence principle exploited in projectile motion.

A projectile is any object launched with initial velocity $\vec{v}_0$ and then subject only to the constant downward free-fall acceleration $\vec{g}$. Air resistance is neglected.

This is not obvious, but is directly verifiable: drop a ball and simultaneously launch another ball horizontally from the same height. Both balls hit the ground at exactly the same instant. The horizontal velocity does not affect the rate of vertical fall.

We decompose the initial velocity into horizontal and vertical components using the launch angle $\theta_0$ above horizontal:

Horizontal: no acceleration, so $v_x = v_{0x}$ throughout. Vertical: free fall with $a_y = -g$. Time $t$ is the single shared variable linking both motions.

Starting at $(x_0, y_0)$ with speed $v_0$ at angle $\theta_0$, the full kinematic equations for each axis are:

At the peak $v_y = 0$. The velocity–position equation gives the maximum height above the launch point:

Speed at any point: $v = \sqrt{v_x^2 + v_y^2}$. At the peak, speed is a minimum: $v_{\text{peak}} = v_0\cos\theta_0$. At launch and landing it equals $v_0$ (for level ground).

Eliminating $t$ between the position equations yields the path equation — the trajectory. Solving $x - x_0 = (v_0\cos\theta_0)t$ for $t$ and substituting into the $y$ equation:

This is $y = ax + bx^2$ — a parabola. Every projectile (neglecting air) follows a parabolic path. When the projectile returns to its launch height, the total horizontal distance is the range:

The range formula applies only when landing height equals launch height. For a cliff, slope, or any non-level geometry, return to the basic position equations and solve for when $y$ reaches the landing elevation.

A particle in uniform circular motion moves at constant speed $v$ on a circular path of radius $r$. Although the speed is constant, the direction of $\vec{v}$ changes continuously — so there is an acceleration.

This acceleration is directed radially inward, toward the center of the circle — it is called centripetal (center-seeking) acceleration:

The time for one complete revolution is the period:

This can be derived directly: if $\vec{r}(t) = r\cos(\omega t)\,\hat{i} + r\sin(\omega t)\,\hat{j}$, differentiating twice gives $\vec{a} = -\omega^2\vec{r}$ — magnitude $\omega^2 r = v^2/r$, directed opposite to $\vec{r}$, i.e., toward the center.

The velocity of a particle is not absolute — it depends on the reference frame, the body to which we attach our coordinate axes. Two observers in frames moving at constant velocity relative to each other will measure different velocities for the same particle.

If frame B moves at constant velocity $\vec{v}_{BA}$ relative to frame A, the velocities of particle P in each frame satisfy:

The subscript notation chains: read "PA" as "P measured in A," and note that the B labels cancel — PB + BA = PA. The rule extends to three or more frames: $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$.

Relative motion is essential in navigation (crosswind corrections, river crossings), collision analysis, and any situation where motion is observed from a moving platform such as a ship, train, or aircraft.