8.5PureStretch

Modelling with Parametric Equations

Parametric equations model motion where the parameter is usually time. They let you describe an object’s horizontal and vertical position separately, then answer questions about range, height and timing.

26 min Video by Zeeshan Zamurred Parametric Equations

Edexcel A level Maths: 8.5 Modelling with Parametric EquationsWatch the full walkthrough before the notes below.

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What you'll be able to do

Interpret t as time in a model
Find position at a given time
Find when an object reaches a height or position
Find greatest height and horizontal range

Time as the parameter

With $x$ and $y$ both functions of time $t$ , substitute a time to get the position, or set $x$ or $y$ to a value and solve for $t$ to find when an event happens.

10 t - 5 t^{2} = 0 \Rightarrow 5 t (2 - t) = 0

t = 0

(start) or

t = 2

AnswerIt lands at

t = 2

s, at

x = 40

Tip — Set y = 0 for landing/ground level; the t = 0 root is the launch.

Greatest height

The greatest height occurs when the vertical component is at its maximum — found by symmetry of the y(t) parabola or by differentiating $y$ with respect to $t$ and setting it to zero.

Formula recap

y = 0 \Rightarrow landing time

Ground level.

\frac{d y}{d t} = 0 \Rightarrow greatest height

Vertical maximum.

Common mistakes to avoid

Taking t = 0 as the landing time.

t = 0 is the launch; the other root is the landing.

Using x to find greatest height.

Greatest height comes from the y-equation.

Key takeaways

The parameter is usually time.
Set y = 0 for landing; the non-zero root is the time of flight.
Greatest height from the maximum of y(t).

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