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Trains, Speed, and Tunnels

Date: 09/10/98 at 19:16:05
From: Matthew
Subject: Trains and Tunnels

A freight train is 500m long. It passes through a tunnel 2000m long. 
Sixty seconds elapse between the time the last car enters and the time 
the engine exits the tunnel. How fast is the train traveling? 

I would probably be able to figure this out if I knew the time when the 
engine enters and exits the tunnel, but I don't know how to calculate 
the speed the way the question says. I've tried drawing a picture on 
graph paper, but it doesn't help. I hope you can help me. Thank you.

Date: 09/11/98 at 17:00:25
From: Doctor Peterson
Subject: Re: Trains and Tunnels

Hi, Matthew. It is a little tricky, but when you see it right, it's 
not too bad. Let's draw a sort of graph, with the distance through the 
tunnel along the x-axis and time along the y-axis (which I'll draw 
downward for convenience):

      in     2000m     out
    0+====================+--->  <-- at time 0, the last car
     |\   500m     \                 enters the tunnel
     | \            \
     |  \            \
     |   \   train    \
     |    \            \
     |     \            \
     |      \            \
   60|       \            \      <-- at time 60, the engine
     V      last         engine      exits the tunnel

You could also just draw a picture of the train and tunnel at the 
start and end of the 60 seconds, to see where it is:

    ___________________________ time 0
   |500m      tunnel           |

    ___________________________ time 60
   |          tunnel       500m|

At time 0, the engine was 500 m into the tunnel, and at time 60 it was 
at the end. So in those 60 seconds, the engine traveled (2000 - 500) 
meters. Now can you finish the problem?

- Doctor Peterson, The Math Forum
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