Jim and Joe Drive toward Each Other

Date: 12/05/2001 at 19:52:05
From: Katie
Subject: Motion problems

Jim and Joe leave their homes at the same time and drive toward each 
other. Jim drives at 60 miles per hour, while Joe drives at 30 miles 
per hour. They pass each other in 10 minutes. How far apart were Jim 
and Joe when they started? 

I got 1/4 mile but I'm not sure where to start. I think I have to use 
the formula distance = rate multiplied by time. Please help!

Date: 12/06/2001 at 09:29:49
From: Doctor Ian
Subject: Re: Motion problems

Hi Katie,

Sometimes the simplest way to solve a problem is see if you can turn 
it into a simpler problem.  

We have Jim and Joe traveling towards each other at 60 mph and 30 mph 
respectively. Suppose that Jim is traveling at 61 mph, and Joe is 
traveling at 29 mph. They would take the same amount of time to meet, 
right?  

Similarly for 62 and 28 mph, 63 and 27 mph... In fact, if Joe stayed 
where he was (0 mph) and Jim traveled at 90 mph, it's really the same 
problem. 

To see why this is the case, imagine that Jim and Joe are on a very 
long railroad car, which is traveling opposite Joe's direction, at 
Joe's speed.  

       Jim --> 60 mph*       30 mph* <-- Joe
     ========================================  --> 30 mph**
     oooo                                oooo

                                          0
                                         /|\
                                         / \
                                       observer

        * relative to the railroad car
       ** relative to the observer

A guy standing still as the train rushed past would see Joe standing 
still relative to him, and would see Jim rushing toward Joe at 90 mph.

So now we have a simpler problem: How far would someone traveling at 
90 mph go in 10 minutes?

It might be helpful to know that 60 mph is the same as 1 mile per 
minute; so 90 mph would be the same as 1 1/2 miles per minute.  

Can you take it from here?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/