# Two Trains and a Bee

## Homework Statement

Consider two trains moving in opposite directions on the same track. The trains start simultaneously from two towns, Aville and Bville, separated by a distance d. Each train travels toward each other with constant speed v. A bee is initially located in front of the train in Aville. As the train departs Aville, the bee travels with speed u>v along the track towards Bville. When it encounters the second train, it instantaneously reverses direction until it encounters the first train, then it reverses again, etc. The bee continues flying between the two trains until it is crushed between the trains impacting each other. The purpose of this problem is to compute the total distance flown by the bee until it is crushed. Assume that the bee is faster than the trains.

Find an expression for the distance d_n covered by the bee after its nth encounter with a train. Define d_0 as the distance traveled during the first flight from Aville towards the train near Bville, d_1 the distance traveled by the bee during the first trip from the Bville train to the Aville train, etc. Sum the resulting series to get the final answer.

I don't know how to proceed with this exercise. Could someone help me please?

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jedishrfu
Mentor
This is a famous story where a mathematician presented this problem to Von Neumann and he figured it out immediately. The mathematician was impresed and said most people I know solve this problem by summing over the infinite series and Von Neumann responded "What do you mean? Thats what I did!"

try solving it step by step and then see if you can find a pattern to the steps and show us your work.

the first step is the bee starts its flight so when will it touch the oncoming train?

This is basically d=ut + vt= (u+v)t right? Then solve for t.

In the next step d is smaller what is the new d value?

Draw a detailed to scale picture (like on graph paper to see the flight of the bee)

Repeat...

and as inspiration listen to this song by yuja wang:

Last edited by a moderator:
phinds
Gold Member
2019 Award
This is a VERY common type problem and the standard confusion is that one tries to get a sum of a series ... WRONG way to go about it.

Just figure out how far the trains travel at what speed and that gives you the time that they travel. Obviously the bee travels for the same amount of time.

Do you see how to proceed from there?

Let d_0: distance traveled during the first flight from Aville toward the train near Bville.

d_0=ut t:some time

d': distance each train travels on the first flight

d'=vt

We have d_0+d'=d

d_0/d'=(ut)/(vt)

<=> d'/d_0=v/u
<=> d'=(v/u)d_0

d_0+(v/u)d_0=d <=> d_0=(u/(u+v))d

d_0-d'
= 2d_0-(d_0+d')
=d(u/(u+v))d-d
=((2u-u-v)/(u+v))d
=((u-v)/(u+v))d

Thus, the distance has been "shrunk" by a factor of 1/5.

Therefore:

d(Total)= (u/(u+v))d*(1+(1/5)+(1/5^2)+...+(1/5^n))

= (u/(u+v))d*(1/(1-(1/5)))

= (5/4)(u/(u+v))d

Is that correct ?

jedishrfu
Mentor
This is a VERY common type problem and the standard confusion is that one tries to get a sum of a series ... WRONG way to go about it.

Just figure out how far the trains travel at what speed and that gives you the time that they travel. Obviously the bee travels for the same amount of time.

Do you see how to proceed from there?
I was going to say the same thing until I saw the nth-step requirement which means the problem is interested in the infinite summation terms.

I am sorry if it isn't so clear. How do we use LaTeX on this forum ?

phinds
Gold Member
2019 Award
I was going to say the same thing until I saw the nth-step requirement which means the problem is interested in the infinite summation terms.
Ah ... I missed that. Just assumed (dumb move) that it was the standard problem.

jedishrfu
Mentor
d(Total)= (5/4)(u/(u+v))d

I do not have any values for u,v and d so..

jedishrfu
Mentor
d(Total)= (5/4)(u/(u+v))d

I do not have any values for u,v and d so..
okay take the values for distance and speed from the primepuzzle link I gave you and plug them in as a sanity check.