Modelling with Parametric Equations Revision Notes for AQA A-Level Mathematics

9.2.3 Modelling with Parametric Equations

Modelling with parametric equations involves representing real-world scenarios where the relationships between variables are best described using a third parameter. This method is particularly useful in situations where the motion or path of an object is considered over time, or when dealing with curves that are difficult to describe with a single function in Cartesian coordinates.

1. Understanding Parametric Equations

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Parametric equations define both $x$ and $y$ coordinates as functions of a third variable, often denoted as $t$ (which could represent time, angle, or another variable):

$x = f(t), \quad y = g(t)$

Here, $t$ is the parameter, and as $t$ changes, the coordinates $x(t), y(t)$ trace out a curve in the plane.

2. Applications and Modelling Scenarios

a) Projectile Motion

Projectile motion is a classic example where parametric equations are used to model the path of an object under the influence of gravity.

Scenario: A projectile is launched with an initial velocity $v_0$ at an angle $\theta$ to the horizontal. The goal is to model its trajectory.

Equations:

Horizontal Motion:

$x(t) = v_0 \cos(\theta) \cdot t$

This describes the horizontal distance travelled over time $t$ , where $v_0 \cos(\theta)$ is the horizontal component of the initial velocity.

Vertical Motion:

$y(t) = v_0 \sin(\theta) \cdot t - \frac{1}{2}gt^2$

This describes the height over time, where $v_0 \sin(\theta)$ is the vertical component of the initial velocity, and $g$ is the acceleration due to gravity. Interpretation:

$x(t)$ and $y(t)$ together define the position of the projectile at any time $t$ .
The resulting curve is a parabola, and the maximum height and range can be found by analysing these equations.

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📑Example Problem: Find the maximum height reached by a projectile launched with an initial velocity of 20 m/s at an angle of 30°.

Solution:

Horizontal component: $x(t) = 20 \cos(30^\circ) \cdot t = 10\sqrt{3} \cdot t$
Vertical component: $y(t) = 20 \sin(30^\circ) \cdot t - \frac{1}{2} \times 9.8 \times t^2 = 10t - 4.9t^2$ To find the maximum height, set $\frac{dy}{dt} = 0$ and solve for $t$ :

$\frac{dy}{dt} = 10 - 9.8t = 0 \quad \Rightarrow \quad t = \frac{10}{9.8} \approx :highlight[1.02 seconds]$

Substitute $t = 1.02$ into $y(t)$ :

$y(1.02) = 10(1.02) - 4.9(1.02)^2 \approx :highlight[5.10 metres]$

The maximum height reached by the projectile is approximately 5.10 metres.

b) Circular Motion

Parametric equations are also ideal for modelling circular or elliptical motion, which is common in physics and engineering.

Scenario: A point moves in a circular path with radius $r$ , centred at the origin, as time progresses.

Equations:

Horizontal Component:

$x(t) = r \cos(\omega t)$

Here, $\omega$ is the angular velocity.

Vertical Component:

$y(t) = r \sin(\omega t)$

Interpretation:

$x(t)$ and $y(t)$ together describe the coordinates of a point moving in a circle.
The parameter $t$ typically represents time, and $\omega$ dictates the speed at which the point travels around the circle.

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📑Example Problem: A point moves in a circle of radius 5 units with an angular velocity of 2 radians per second. Find the position of the point at $t = \frac{\pi}{2}$ seconds.

Solution:

$x(t) = 5 \cos(2t)$
$y(t) = 5 \sin(2t)$ At $t = \frac{\pi}{2}$ :

$x\left(\frac{\pi}{2}\right) = 5 \cos\left(2 \times \frac{\pi}{2}\right) = 5 \cos(\pi) = -5$

$y\left(\frac{\pi}{2}\right) = 5 \sin\left(2 \times \frac{\pi}{2}\right) = 5 \sin(\pi) = 0$

The position of the point at $t = \frac{\pi}{2}$ seconds is (-5, 0).

c) Ellipse

Elliptical motion, such as planetary orbits, can also be modelled with parametric equations.

Scenario: A point moves along an elliptical path where the semi-major axis is $a$ and the semi-minor axis is $b$ .

Equations:

Horizontal Component:

$x(t) = a \cos(t)$

Vertical Component:

$y(t) = b \sin(t)$

Interpretation:

The curve is an ellipse centred at the origin, with axes lengths determined by $a$ and $b$ .

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📑Example Problem: Model the orbit of a planet with a semi-major axis of 10 units and a semi-minor axis of 6 units.

Solution:

$x(t) = 10 \cos(t)$
$y(t) = 6 \sin(t)$ This set of equations models the elliptical orbit of the planet.

3. Summary

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Modelling with parametric equations is a powerful technique for describing motion and curves in various applications:

Projectile motion: Captures the trajectory of an object under gravity.
Circular and elliptical motion: Describes the movement of objects in orbits or rotations.
Complex curves: Parametric equations allow the representation of curves that are difficult or impossible to describe with a single Cartesian equation.

Modelling with Parametric Equations (AQA A-Level Mathematics): Revision Notes

9.2.3 Modelling with Parametric Equations

1. Understanding Parametric Equations

2. Applications and Modelling Scenarios

a) Projectile Motion

b) Circular Motion

c) Ellipse

3. Summary

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