Topic 7 of 8

Parametric equations

Step 14 · Trigonometry and precalculus topics

Describe motion with x and y as functions of time, eliminate the parameter to find the path, and model a projectile's flight.

Position over time

Parametric equations give x and y separately, each as a function of a third variable, usually time t. As t changes, the point (x(t), y(t)) traces out a path.

Example: Making a table

x = 2t and y = t² − 1. Find the points for t = 0, 1, 2 and 3.

t = 0 gives (0, −1), and t = 1 gives (2, 0).
t = 2 gives (4, 3), and t = 3 gives (6, 8).

Answer(0, −1), (2, 0), (4, 3) and (6, 8)

x = 2t, y = t² − 1 for t from 0 to 3: the point moves along a parabola.

Eliminating the parameter

To find the ordinary equation of the path, solve one equation for t and substitute into the other.

Example: A line

Eliminate t from x = t + 1 and y = 3t − 2.

From the first, t = x − 1.
y = 3(x − 1) − 2, which is 3x − 5.

Answery = 3x − 5

Example: A parabola

Eliminate t from x = 2t and y = t² − 1.

t = x ÷ 2.
y = (x ÷ 2)² − 1.

Answer $y = \frac{x^{2}}{4} - 1$

Example: A circle

Eliminate t from x = 3 cos t and y = 3 sin t.

cos t = x ÷ 3 and sin t = y ÷ 3.
cos²t + sin²t = 1 gives x² ÷ 9 + y² ÷ 9 = 1.

Answerx² + y² = 9

Projectile motion

A ball launched at speed v and angle θ moves sideways at a steady v cos θ, while gravity pulls its height down.

x = (v\cos\theta)\,t \qquad y = h + (v\sin\theta)\,t - 4.9t^{2}

Here x and y are in meters, t is in seconds, and h is the starting height.

Example: Time in the air

A ball is kicked from the ground at 20 m/s at 30°. When does it land?

Up speed: 20 sin 30° = 10. Across speed: 20 cos 30°, about 17.3.
y = 10t − 4.9t² is 0 again when t = 10 ÷ 4.9, about 2.04 s.

AnswerAfter about 2.04 seconds

Example: How far

How far away does it land?

x = 17.3 × 2.04, about 35.3.

AnswerAbout 35.3 m

The ball's height against time: highest, about 5.1 m, at 1.02 s, and back to the ground at about 2.04 s.

Try it yourself

Work each one on paper first, then open the answer to check.

x = t − 3 and y = 2t. Where is the point at t = 4?
Show answer
(1, 8)
Eliminate t from x = t + 2 and y = t².
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y = (x − 2)²
Eliminate t from x = 5 cos t and y = 5 sin t.
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x² + y² = 25
x = 4t and y = 8 − t. When and where does the path cross the x-axis?
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At t = 8, at the point (32, 0)
A ball is thrown straight out at 10 m/s from a height of 20 m. Its height is y = 20 − 4.9t². When does it land, to two decimal places?
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About 2.02 s
In the same throw, x = 10t. How far does it travel before landing?
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About 20.2 m