A runner and his trainer are standing together on a circular track of radius . When the trainer gives a signal, the runner starts to run around the track at a speed of . How fast is the distance between the runner and the trainer increasing when the runner has run a quarter of the way around the track?

Question

A runner and his trainer are standing together on a circular track of radius  . When the trainer gives a signal, the runner starts to run around the track at a speed of  . How fast is the distance between the runner and the trainer increasing when the runner has run a quarter of the way around the track?

Yihan · Accepted Answer

The situation above can be illustrated as shown below:

Let  distance between the trainer and the runner at any instant on the track
We can mathematically represent the positions of the runner and the trainer using Cosine Law.
  
The radius here is always constant so upon differentiating both sides with respect to the time  , we get
    
Or
    
 
When the runner has run a quarter away around the track,  . At that instance, we can also solve for  using Cosine Law,
  
  
 
Meanwhile, the speed of the runner has to be converted to radians per second based on our equation.
We can use the circumference of the track to derive the conversion factor.
  
This means that for every one revolution or   , the runner has run a total of  .
So that,
  
 
We can now solve for the unknown as follows.
  
 
Therefore, the distance between the runner and the trainer is increasing at a rate of  .

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