Question

Find parametric equations that describe the circular path of the following object. Assume​ (x,y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time given in the description. A​ go-cart moves clockwise with constant speed around a circular track of radius 600 ​m, completing one lap in 1.8 min.

Answer 1

(x, y) = (600cos(10πt/9), -600sin(10πt/9))

Explanation:

The usual translation between rectangular and polar coordinates is ...

• x = r·cos(θ)
• y = r·sin(θ)

Here, the radius is constant at r = 600 m, and the angle changes linearly with time. If we assume the initial angle is 0, then it is -2π radians (one full turn clockwise) at t=1.8 minutes. The relationship between θ and t will be given by ...

θ = (-2π)(t/1.8) = -10πt/9

Using these values for r and θ, we get the parametric equations ...

• x = 600·cos(-10πt/9)
• y = 600·sin(-10πt/9)

We can take advantage of the fact that cosine is an even function, so cos(-θ) = cos(θ), and that sine is an odd function, so sin(-θ) = -sin(θ). This lets us write the equations as ...

(x, y) = (600·cos(10πt/9), -600·sin(10πt/9))