Lenth of a train

[Rassis]

I have just seen this problem in an article dedicated to Maths in a newspaper. I found it not too difficult and quite interesting. I couldn’t resist posting it in this forum. Please enjoy.

Two mates, Alexander and Michael, wait at the train station for their scheduled train. At a certain moment a merchant train entered the station without stopping. To help time goes by, Alexander and Michael decide to calculate the length of the merchant train that is passing at constant velocity. They wait until the front of the train reaches the point where they stand and then they start walking at the same velocity. Alexander walks towards the same direction of the train and Michael towards the opposite direction. Each one of them stops as the rear of the train reaches each one’s position. Alexander walked a distance of 45 meters and Michael 30 meters.

What is the length of the train?

[krtxmrtz]

Let x₀ = 0 and t₀ = 0 the point and moment they start walking, x_A the distance walked by Alexander and t_A the time it takes him to walk it. Let x_M and t_M the analogous values for Michel. If we take as the positive x axis that along which both Alexander and the train move, then x_M will be negative. Let also v_T the train velocity and v (and -v) the walking friends' velocities, and call L the train length we want to calculate.

Then:

x_A = vt_A
x_B = -vt_B
x_A + L = v_Tt_A
x_M + L = -v_Tt_M

We have 4 equations with 4 unknowns, so it's not difficult to solve and arrive at:
L = -2x_M / (1 + x_M / x_A) = 180 m

[Rassis]

Krtxmrtz,

I said it wouldn’t be difficult. Your answer is, of course, correct and the explanation is quite algebraic oriented and elegant. Thanks for taking the time.

Rui

[krtxmrtz]

Krtxmrtz,
...Thanks for taking the time.

My pleasure!