Applications of Parametric Equations

Question

At noon a runner starts 3miles north of his house. At 1:00npm he is at a point that is 1 mile west and 1 mile north of his house.

A) how far has he ran in his first hour ?

B) if he continues at this pace where will he be in relation to his house at 3:00pm

C) what function can I write that will give the runner's position at any given time

Answer 1

First set up a coordinate system with the runner's house at (x,y) = (0,0) and noon corresponding to time t = 0.

The runner's initial position is then (0,3)

A)

The runner's position at time t=1 hour is (-1,1).

You can use the distance formula with (0,3) and (-1,1) to find how far he has ran.

B) & C)

His velocity (distance/time) in the x direction is v_x = (-1 - 0) / 1 = -1

His velocity (distance/time) in the y direction is v_y = (1 - 3) / 1 = -2

The parametric equations for motion follow this form:

x(t) = x₀ + v_xt

y(t) = y₀ + v_yt

The equations for his motion are then:

x(t) = -t

y(t) = 3 - 2t

Substitute t = 3 for x(t) and y(t) to find his position at 3pm.

x(3) = -3

y(3) = -3