Question

Jeremy's coach makes him run clockwise around a circular track with radius of 50 meters. Jeremy manages to maintain a constant speed around the track. He takes 48 seconds to finish one lap of the track. From his starting point, it takes him 12 seconds to reach the northernmost point of the track. Answer the following questions below assuming that the center of the track at the origin and the northernmost point is on the y-axis.

a. Give Jeremy's coordinates at his starting point.
b. Give Jeremy's coordinates when he has been running for 4 seconds.
c. Give Jeremy's coordinates when he has been running for 32 seconds.

Answer 1

Coordinates of the starting point ( -50 ; 0 )

Coordinates 4 seconds later Q ( - 25*√3 ; 25 )

Coordinates 32 seconds later R ( 25 ; - 25*√3 )

Explanation:

a) If Jeremy takes 48 seconds fr a lap then

The length of the lap ( length of the circle ) is:

L = 2*π*r ⇒ L = 100*π

If the time for one lap was 48 seconds at a constant speed, the speed was

v = 100*π / 48 [m/s]

v = 6,54 m/s

12 seconds is 1/4 0f 48 in that time he (she) reach the northernmost point, then he(she) necessarily started on the negative side of the x-axis the coordinates at this point are

( -50 ; 0 )

b) 4 seconds later, at v = 6,54 m/s by rule of three

In 12 seconds 90⁰

In 4 seconds (the third part ) x ??

x = 30⁰

sin 30⁰ = 1/2

cos 30⁰ = (√3)/2

And coordinates for the point are

sin 30⁰ = 1/2 = y/50 ⇒ y = 25

cos 30⁰ = (√3)/2 = x / 50 ⇒ x = 25*√3

coordinates of the point 4 seconds later

Q ( - 25*√3 ; 25 ) she (he) is in the negative part of x-axis

c) 32 seconds later

32 is 12 + 12 + 8

Then she (he) is 8 seconds below the positive side of x-axis

8 is 2/3 of 12 ( negative 60⁰ )

sin 60⁰ = (√3)/2 = y/50 y = 25*√3 negative

cos 60⁰ = 1/2 * 50 = X/50 x = 25

Coordinates of the point R

R ( 25 ; - 25*√3 )