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### 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

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
```
Associated Topics:
High School Basic Algebra
High School Coordinate Plane Geometry
Middle School Algebra
Middle School Word Problems

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