Measuring the Length of a Moving Train

Date: 08/12/2003 at 13:36:48
From: Huey D.
Subject: How do I measure the length of a moving train?

I was driving in the opposite direction from a moving train that had 
many cars behind the locomotive. It took me approximately 2-3 minutes 
from the time my car "met" the front of the train until it passed me 
going the opposite direction. My speed was about 40 miles an hour. 

Is it possible to estimate the length of this moving train without 
actually measuring when it is not moving?

Date: 08/12/2003 at 18:02:42
From: Doctor Warren
Subject: Re: How do I measure the length of a moving train?

Hi Huey,

Thanks for writing to Dr. Math.

The speedometer in your car measures your velocity with respect to the
ground; I'll call this velocity v_cg (v subscript c,g), meaning
"velocity of the car with respect to (wrt) the ground."  It's
important in physics problems that you always remember how you define
your variables. You could select to measure the car's velocity with
respect to the ground, or with respect to an airliner flying overhead,
or with respect to the Andromeda galaxy. Depending upon which 
reference you choose, your car's velocity will be very different.  So
don't forget that v_cg is specifically relative to the ground.

Now, let's define v_tg to be the velocity of the train with respect
to the ground.

What is your car's velocity with respect to the train? It's the sum
of the car's velocity wrt the ground and the train's velocity wrt the
ground (assuming the car and train are travelling in opposite 
directions):

   v_ct = v_cg + v_tg

Ask yourself: does this make sense?  If your car is headed down the
road at 60 mph wrt the ground, and the train is headed the other way
at 60 mph wrt the ground, then the velocity between you and the train
is 60 + 60 = 120 mph. It does indeed make sense.

Now, you know that distance = rate * time. In other words, you can
measure the length of the train if you know a) how fast it's moving
with respect to you, and b) how much time it takes you to pass the
whole thing from start to finish. Read that sentence over again, and
notice the important words "with respect to you."  The number you need
to know to measure the length of the train is its velocity with
respect to you. This value, v_ct, is the sum of v_cg and v_tg. Your
car's speedometer gives you v_cg, but unfortunately, you don't know
v_tg at all.

The overall formula for the measurement is:

   length of train = total velocity / time
                   = v_ct / time
                   = (v_cg + v_tg) / time

Since you don't know v_tg, you can't calculate the length of the
train.  Ask yourself: does this make sense?

Let me know if you have any more questions!

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