Distance of to people running away from each other, derivates

Question

Pam is seated 300m away from the center of a circular track of radius 100m and is watching John run laps at a speed of 5m/s. How fast is the distance between Pam and John changing when John is approaching Pam and the distance between them is 250m.

Answer 1

This Desmos graph helps to visualize the scenario:

desmos.com/calculator/lthklf0gwe

Be sure to collapse the lefthand side by clicking the << icon at top of lefthand panel.

One of the keys to this problem is realizing that the 5 m/s is linear velocity (speed running around the circle). This must be converted to angular velocity because the central angle at center of circle is used to calculate the distance from John to Pam at some point in time.

Angular Velocity equals linear / radius of circle, so dθ/dt = 5/100 = .05 or 1/20 radians per second.

Set up a rectangular coordinate system with center of circular track at C(0,0) and Pam sitting at P(300,0). John is moving around the circle x² + y² = 100² with angular velocity .05 rad/sec. Let J be the point at some point in time where John is located on that circle. Then segment JP is the distance from John to Pam. Let the measure of that segment be represented by s. The segment CJ is a radius of the circle with measure 100. Segment CP is joins center of track to Pam. Consider ΔCJP with side lengths 100, 300, s and angle θ included between the 100 and 300 legs. Then by the Law of Cosines:

s² = 100² + 300² - 2(100)(300)cosθ = 100000 - 60000cosθ.

Differentiating with respect to t:

2s (ds/dt) = 60000sinθ(dθ/dt)

ds/dt = [(30000sinθ)/s] (dθ/dt)

We know dθ/dt, but need to determine sinθ in order to determine ds/dt when s = 250.

We can determine the cosθ from the Law of Cosines.

250² = 100² + 300² - 2(100)(300)cosθ

cosθ = [100² + 300² - 250²]/ 60000 = 5/8

Using sin²θ + cos²θ = 1, solve for sinθ.

sinθ = ±√[1 - (5/8)²] = ±√39/8. Since we want John moving towards Pam, place θ in the fourth quadrant and use the negative value for sinθ.

Finally ds/dt = [30000(-√39/8)(.05)]/250 ≈ -4.69 meters/sec.