Question

Georgia is roller skating clockwise on a circular track. F rom her starting point, it takes her 5 seconds to reach the southernmost point of the track, an d 21 seconds to reach the westernmost point. Georgia skates at a constant 3.2 meters p er second. (a) Express Georgia’s coordinates as functions of time, usi ng the center of the track as the origin. (b) After skating for 17 minutes, how far (in a straight line) will she be from the north- ernmost point?

Answer 1

Refer to the figure shown below.

Define
A =  Georgia's starting position (at time t = 0)
B = Georgia's position after 5 seconds
C = Georgia's position after 21 seconds.
ω = angular velocity, rad/s
θ₁ =  initial central angle (measured from the southern location)
r = radius of the circle, m

Between B and C, Georgia sweeps a central angle of π/2 in (21-5) = 16 seconds.
Her angular velocity is
ω = (π/2 rad)/(16 s) = π/32 = 0.0982 rad/s

Because the trip from A to B takes 5 seconds, therefore
θ₁ = (π/32 rad/s)*(5 s) = (5π)/32 = 0.491  rad

Georgia's linear speed is v = 3.2 m/s. She travels from A to B in 5 seconds.
The arc length is
S = rθ₁ = vt
(r m)*( (5π)/32 rad) = (3.2 m/s)*(5 s)
r = 102.4/π = 32.595 m

The coordinates of A are (r sinθ₁, -rcos θ₁) = (15.365, -28.746) m
The coordinates of B are (0, -32.595) m
The coordinates of C are (-32.595, 0) m

Relative to the positive x-axis (measured clockwise), Georgia's position as function of time is
θ(t) = π/2 - θ₁ + ωt
      = 1.08 + 0.0982t  rad
In rectangular coordinates,
x(t) = 32.595*cosθ m
y(t) = -32.595*sinθ m

Part (a)
Georgia's position as function of time is
x(t) = 32.595 cos θ(t)
y(t) = -32.595 sin θ(t)
where
θ(t) = 1.08 + 0.0982t rad
t = time, s

Part (b)
After 17 minutes (=17*60 = 1020 s), Georgia's position is determined by
θ = 1.08 + 0.0982*1020 = 101.244 rad 
This is equal to 
101.244/(2π) = 16.1135 revolutions
                     = 0.1135 radians clockwise from the x-axis.
Relative to the north, her position is
x = 32.595*cos(0.1135) = 32.385 m
y = -32.595*sin(0.1135) = -6.692 m
The northern coordinate is (0, 32.595)
The length of the straight line joining her position to the northern point is given by
d² = 32.385² + (32.385 + 6.692)² = 2605.9
 d = 51.05 m

Answer:
51.05 m in a straight line joining her position to the northern-most point.