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 −2π≤t≤2π as:
xy=0.5t−4sint=1−3cost
A sketch of the curve is shown. Find
a) The length of the straight line AB.
b) The depth of the ramp.
First, find the coordinates of some points on the curve. Use t=−2π and 2π 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:
| −2π | | |
| 4−4π | | 4π−4 |
| | | |
a) From the table, it can be seen that point A has coordinates (4π−4,1) and B has coordinates (4−4π,1). Therefore the distance AB is the difference in the x coordinates:
ABAB=(4−4π)−(4π−4)=8−2π
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)=3 m.