Two trains (1 and 2), each having a speed u are initially ( i.e. t = 0) a distan

Question

Two trains (1 and 2), each having a speed u are initially (i.e. t = 0) a distance d apart and are headed towards each other on the same track. A bird that can fly at a speed v (where v > u) flies from the front of train 1 directly towards train 2. On reaching train 2, the bird turns around and immediately flies back to train 1, and so forth. The goal of this problem is to calculate the total distance the bird flies before the trains crash into each other.

(a) Calculate the time at which the trains crash.
(b) Calculate the total distance the bird flies before the trains crash.
(c) Sketch the position of the two trains and of the bird as a function of time.

Explanation / Answer

initial distance = d

with respect to train 1 the velocity of bird = v-u

with respect to train 2 the velocity of bird = v+u

so it travels for t= d/(u+v)

by that time each train travelled u*d/(u+v) so the distance becomes d- 2ud/(u+v) =( v-u)d/(u+v)

now the same case is repeated with distance (v-u)d/(u+v) until the trains crash at t= d/2u

1. The trains crash at t= d/2u

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