Modelling with parametric equations

In a nutshell

Parametric equations can be used to model many real life situations. There are many applications and some may relate to mechanics problems, including projectile motion.

Real life models

Mechanics problems can be modelled using parametric equations. For example, the equations for $x$ and $y$ can represent the displacement in different directions and the parameter can represent time.

Example

The motion on a skateboard ramp can be modelled by the parametric equations for $-\frac {\pi} {2} \le t \le \frac {\pi} {2}$ as:

$\begin{aligned}x&=0.5t - 4\sin t \\y&=1-3 \cos t\end{aligned}$ 

A sketch of the curve is shown. Find

a) The length of the straight line $AB$ .

b) The depth of the ramp.

Maths; Parametric equations; KS5 Year 13; Modelling with parametric equations

First, find the coordinates of some points on the curve. Use $t=- \dfrac{\pi}{2}$ and $\dfrac{\pi}{2}$ as these values are given as the limits for the range of $t$ . The curve also intersects the $y$ -axis, so find where $x=0$ . Use a table of values:

$t$	$-\dfrac {\pi}{2}$	$0$	$\dfrac{\pi}{2}$
$x$	$4- \dfrac{\pi}{4}$	$0$	$\dfrac {\pi}{4} - 4$
$y$	$1$	$-2$	$1$

a) From the table, it can be seen that point $A$ has coordinates $\Big( \frac{\pi}{4}- 4, 1\Big)$ and $B$ has coordinates $\Big( 4-\frac{\pi}{4}, 1\Big)$ . Therefore the distance $AB$ is the difference in the $x$ coordinates:

$\begin{aligned}AB &= \Big( 4-\dfrac{\pi}{4}\Big) - \Big(\dfrac{\pi}{4}- 4 \Big) \\\\AB&=\underline{8 - \dfrac{\pi}{2}}\\\end{aligned}$ 

b) The depth of the ramp is given by the difference in $y$ -coordinates. Points $A$ and $B$ have $y$ -coordinate $1$ and the lowest point is on the $y$ -axis, where $x=0$ . At this point, $y=-2$ .

Therefore, the depth is $1-(-2) = \underline{3 \text{ m}}$ .