Ball Thrown between Trains

Date: 10/15/2003 at 03:09:36
From: Juily
Subject: trains

Two trains A and B each of length 100m travel in opposite directions 
in parallel tracks.  The speeds are 20m/s and 30m/s respectively.  A 
boy sitting in the front end of train A throws a ball to a boy sitting 
in the front end of train B when they are at the closest distance.  
The speed of the ball is 2m/s.  The ball, instead of reaching the boy,
hits the rear end of the train.  Find the distance between the 
parallel tracks. 

I am confused as to whether I should take the vector component of the 
speed of train A and the ball.  I do not have an answer to the 
question to verify my answer with.

Date: 10/15/2003 at 07:22:23
From: Doctor Luis
Subject: Re: trains

Hi Juily,

The ball will obtain a parallel component equal to 20 m/s, which is 
the speed of the train A, and a perpendicular component equal to 2 
m/s, which is the speed at which the boy threw the ball.

Now, when I say parallel and perpendicular components, I mean relative 
to the track.  Here's a diagram representing the situation at t = 0, 
the time when the boy threw the ball

  

Let the position of the ball at time t = 0 be x = 0, y = 0.  That is, 
define the origin at that point in time.

What is the position of the ball at time t = T secs, when the ball 
hits the back of train B?  Well, the x-component traveled at +20m/s, 
and the y-component traveled at +2m/s, so we have

   x = (20 m/s)T
   y = (2 m/s)T

But at time t = T, we know that the ball hit the back of train B. What
does this mean?  Well, the ball made it to the other track, so this 
means that y = H, the separation between the tracks.  Also, we
know that after T seconds, the front of train B traveled 30T meters to 
the left.  This means the back of the train, which started at x = +L 
(since the train is L meters long), must now be at x = L - 30T.

Since these are the coordinates of the ball, we must satisfy the 
equations

  (20 m/s)T = L - (30 m/s)T

   (2 m/s)T = H

Using the first equation, you can solve for T using the train length 
L = 100 m.  And once you know T, you can finally find H, what the 
problem asked for.

I hope this helped!  Let us know if you have any more questions.

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

Date: 10/15/2003 at 08:32:25
From: Juily
Subject: Thank you (trains)

Hi Dr. Luis,

Thank you for your time and for such a quick response.  I think that 
all of you are doing a great job.

Looking forward to asking you more questions,
Juily