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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 Parametric Equations
Edexcel A level Maths: 8.5 Modelling with Parametric EquationsCurated by Kepler, not produced by us. Watch it 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
1

Time as the parameter

With xx and yy both functions of time tt, substitute a time to get the position, or set xx or yy to a value and solve for tt to find when an event happens.

Example 1A ball has x=20tx = 20t, y=10t−5t2y = 10t - 5t^2. When does it land (y=0y = 0)?
110t−5t2=0⇒5t(2−t)=010t - 5t^2 = 0 \Rightarrow 5t(2 - t) = 0.
2t=0t = 0 (start) or t=2t = 2.
Answer
It lands at t=2t = 2 s, at x=40x = 40.

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

2

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 yy with respect to tt and setting it to zero.

Formula recap

y=0⇒landing timey = 0 \Rightarrow \text{landing time}
Ground level.
dydt=0⇒greatest height\tfrac{dy}{dt} = 0 \Rightarrow \text{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).

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. In a motion model, the parameter t usually represents:

    • time
    • distance
    • angle
    • speed
    Show answer and explanation

    Answer: time

    Position vs time.

  2. For y=10t−5t2y = 10t - 5t^2, landing (y=0y=0) is at t =

    • 22
    • 00
    • 55
    • 1010
    Show answer and explanation

    Answer: 22

    5t(2−t)=0, non-zero root.

  3. With x=20tx = 20t, the horizontal range at landing (t=2t=2) is:

    • 4040
    • 2020
    • 1010
    • 8080
    Show answer and explanation

    Answer: 4040

    20 × 2.

  4. Greatest height is found from:

    • the y-equation
    • the x-equation
    • the sum x + y
    • neither
    Show answer and explanation

    Answer: the y-equation

    Vertical component.

  5. The root t = 0 in a projectile model represents:

    • the launch
    • the landing
    • the peak
    • the range
    Show answer and explanation

    Answer: the launch

    Starting instant.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.