Traveling Buses

Date: 06/01/97 at 03:26:21
From: Richard Ramsey
Subject: Uniform motion both equal and unequal distances

Here is my problem.

A southbound bus left Fort Walton Beach at 9 AM.  Two hours later a 
northbound bus left the same station.  If the buses traveled at the 
same rate and were 352 kilometers apart at 2 PM, find the rate of the 
buses.

Here is a sample problem where they travel unequal distances - please 
help me!

At 8 PM Achilles left camp and headed south at 20 kilometers per hour.  
At 10 PM, Patrocles headed south from the same camp.  If Patrocles was 
50 kilometers ahead by 3 AM, what was his speed? 

Date: 06/01/97 at 05:30:27
From: Doctor Mitteldorf
Subject: Re: uniform motion both equal and unequal distances

Dear Richard,
    
The key step in any word problem is to take the words in the problem 
and write them as an equation.  It takes understanding and experience.  
Practice and lots of examples are the only ways I know to get good at 
this.
    
In the first problem, it says the two rates were the same.  We'll let 
that be our variable, and we'll call it v for velocity.  This problem 
happens over the course of 5 hours (9 to 2) and it's divided into two 
parts: in the first 2 hours, there's one bus moving and one standing 
still; in the last 3 hours, both buses are moving but in opposite 
directions.  During the first 2 hours, the separation increases by 
2*v: speed x time = distance. During the last 3 hours, each bus 
covers a distance of 3*v.  We also know that the buses are 352 km 
apart after five hours, so the sum of the distances traveled should be 
equal to 352 km, right? 
    
The equation that comes out of all this is:

     2*v + 3*v + 3*v = 352
                 8*v = 352 
                   v = 352/8 
                   v = 44 km/hr

The second problem sounds similar, but you really have to start over 
again and think it through.  What is it you want to know here?  
Achilles's path is completely known from the start.  He travels for a 
total of 7 hours at a speed of 20 kph, so he goes 140 miles all 
together.  Patrocles goes 50 km further, or 190 km all together.  He 
makes the 190 km trip in only 5 hours, since he started at 10 PM, so 
he must be traveling at 190/5 = 38 kph.

For more about this kind of word problem, see "Two Trains" in the Dr. 
Math FAQ:

  http://mathforum.org/dr.math/faq/faq.two.trains.html   

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/