Motion in a Plane

The last chapter described motion along a single straight line, where a plain $+$ or $-$ sign was enough to capture direction, since only two directions were ever possible. Motion in a plane or in space needs more than a sign: it needs vectors.

A scalar is a quantity with magnitude only, completely specified by a single number and a unit, distance, mass, temperature, time. Scalars combine by ordinary algebra. A vector has both a magnitude and a direction, and combines by a different rule entirely, the triangle law or the parallelogram law of addition, not plain arithmetic. Displacement, velocity, acceleration, and force are all vectors.

Multiplying a vector $\mathbf{A}$ by a positive number $\lambda$ scales its magnitude by $\lambda$ and leaves its direction unchanged. Multiplying by a negative number scales the magnitude by $|\lambda|$ and flips the direction. Multiplying a constant velocity vector by a duration of time, for instance, produces a displacement vector, and the dimensions multiply along with the numbers.

Adding two vectors $\mathbf{A}$ and $\mathbf{B}$ graphically means placing $\mathbf{B}$’s tail at $\mathbf{A}$’s head and drawing the vector from $\mathbf{A}$’s tail to $\mathbf{B}$’s head, the head-to-tail or triangle method. Equivalently, bringing both tails to a common origin and completing a parallelogram gives the sum along its diagonal, the parallelogram method; the two methods always agree. Vector addition is commutative and associative:

A vector added to its own negative gives the null vector $\mathbf{0}$, which has zero magnitude and, since there’s nothing left to point anywhere, no defined direction. Subtraction is just addition of the negative: $\mathbf{A}-\mathbf{B}=\mathbf{A}+(-\mathbf{B})$.

Just as a vector can be built up from two others, it can be taken apart into two component vectors along any pair of non-parallel directions in the same plane. The most useful choice is a rectangular coordinate system with unit vectors $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ along the x- and y-axes, each of magnitude exactly 1 and carrying no dimension of its own, they only specify direction. Any vector $\mathbf{A}$ in the plane can then be written as:

where $\theta$ is the angle $\mathbf{A}$ makes with the x-axis. A component like $A_x$ is a plain number, not a vector, it can come out positive, negative, or zero depending on $\theta$. Given the components instead of the magnitude and angle, the reverse conversion is just Pythagoras and inverse tangent:

The graphical method is intuitive but tedious and only ever as accurate as the drawing. Once every vector is written in component form, addition becomes ordinary arithmetic done twice, once per axis:

The same component-by-component rule extends to any number of vectors, in three dimensions as easily as two, and to subtraction by simply flipping a sign before adding.

Everything from one-dimensional kinematics carries over once position, velocity, and acceleration are written as vectors. The position vector is $\mathbf{r} = x\hat{\mathbf{i}} + y\hat{\mathbf{j}}$, and a displacement over some interval is $\Delta\mathbf{r} = \Delta x\,\hat{\mathbf{i}} + \Delta y\,\hat{\mathbf{j}}$.

Average velocity is displacement divided by the time interval, $\bar{\mathbf{v}} = \Delta\mathbf{r}/\Delta t$, and points along $\Delta\mathbf{r}$. Shrinking $\Delta t$ toward zero turns this into the instantaneous velocity:

As the time interval shrinks, the displacement direction rotates smoothly until it lies exactly along the curve itself, so velocity at any point is always tangent to the path, pointing the way the object is actually moving. This is worth holding onto: in one dimension, velocity and acceleration are always along the same line, but for motion in a plane the angle between them can be anywhere from $0^\circ$ to $180^\circ$.

Acceleration follows the identical pattern one derivative up:

When acceleration $\mathbf{a}$ stays constant, the same equations from straight-line motion apply, just written as vectors, and consequently apply separately and independently to each axis:

In components, this is really two separate one-dimensional problems running at once:

That independence of the two axes is the single most useful idea in this chapter: motion in a plane is nothing but two ordinary one-dimensional motions, along two perpendicular directions, happening at the same time without interfering with each other. Projectile motion, next, is the clearest possible illustration.

An object launched into the air and then left to gravity alone, a ball, a shell, a shot put, is a projectile. Its motion is exactly the superposition described above: horizontally, nothing acts on it (ignoring air resistance), so that component of velocity never changes; vertically, gravity provides constant acceleration $-g$. It was Galileo who first argued, in 1632, that these two components could be treated as independent.

Launching at speed $v_0$ and angle $\theta_0$ from the origin gives:

Eliminating $t$ between the two shows the path is a parabola, and standard bookkeeping on the same two equations gives the time of flight, maximum height, and horizontal range:

For a fixed launch speed, $R$ is largest when $\sin 2\theta_0$ is largest, at $\theta_0=45^\circ$. The full derivation, an interactive simulation, and exactly where the familiar “time up equals time down” and “45° is always best” shortcuts quietly assume equal launch and landing heights, live on the concept page.

An object moving at constant speed around a circle is in uniform circular motion. “Uniform” describes the speed only, the velocity itself is constantly changing direction, tangent to the circle at every instant, and a changing velocity means the object is accelerating even though its speed never changes.

Comparing the velocity vectors at two nearby points and taking the limit as they merge shows that this acceleration always points toward the centre of the circle, which is why it is called centripetal (“centre-seeking”) acceleration, a term Newton proposed. Its magnitude works out to:

A second, equivalent description uses the angular speed $\omega$, the rate at which the radius vector sweeps out angle, $\omega = \Delta\theta/\Delta t$. Since the arc length covered is $R\,\Delta\theta$, the linear speed is $v=\omega R$, and substituting back:

In terms of the time period $T$ (one full revolution) or frequency $\nu = 1/T$: $\omega = 2\pi\nu$, $v = 2\pi R\nu$, and $a_c = 4\pi^2\nu^2 R$. Because the direction of $\mathbf{a}_c$ keeps changing even though its magnitude doesn’t, centripetal acceleration is not a constant vector, and the ordinary constant-acceleration equations from §3.6 do not apply to circular motion at all.

A few of the chapter’s own exercise questions, with original worked solutions.