Modelling with parametric equations

Using parametric equations to model real situations where x and y each depend on time — projectiles and rotating points — reading positions, maxima and ranges from the model, and finding the Cartesian equation of the path.

Pure · Coordinate geometry

Modelling with parametric equations

Why it works

Parametric equations earn their keep in modelling, because in the real world horizontal and vertical position are usually separate functions of time. A single Cartesian equation hides the time information; the parametric form keeps it, so you can answer "where is it?" and "when?" in the same breath. Here the parameter

t

genuinely is time (seconds), and

x

y

are positions (metres).

Reading the model. Most questions are substitution in disguise:

Position at a given time — put the value of $t$ into both equations.
When does it reach the ground / the axis? — set $y = 0$ (or whichever coordinate)

and solve for

t

. Discard any negative

t

if time starts at

0

Greatest height / range — the height $y(t)$ is typically a quadratic in $t$ ;

complete the square (or use the vertex) to find its maximum and the time it occurs. The horizontal range is the

x

-value at the moment it lands.

The path — eliminate $t$ to get the Cartesian equation $y$ in terms of $x$ ,

the actual shape of the trajectory.

A projectile thrown with

x = 20t,\ y = 15t - 5t^2

traces the arc below: it leaves the origin, rises to a greatest height, and lands again when

y = 0

.Rotating points use the circle parametrisation. A point on a wheel of radius

a

with centre

(p, q)

has position

x = p + a\cos t,\ y = q + a\sin t

; its height

y

swings between

q - a

and

q + a

, and its path is the circle

(x - p)^2 + (y - q)^2 = a^2

.

Keep the units and the domain.

t \ge 0

usually (time doesn't run backwards), and an answer like a height or a distance should carry its unit and be physically sensible.