How do I solve this problem?

Two trains move towards one another. The first one has a speed of 30 km/h, and the second one travels at 50 km/h. The initial distance between them is 160 km and they both begin to move at the same time.

A bird resting on the first train is scared when it starts to move, so it begins to fly towards the second train with 70 km/h. Whenever the bird encounters a train, it will change direction towards the other one. How much would the bird have traveled when the two trains met?

This is a physics problem that requires a good look at what we are required to find out. The wrong way to aproach this problem is to calculate when will the bird first met the second train, then when it will met the first train again and so on.

The trick is to find out for how much time will the bird fly. This will be from the moment the two trains start moving, until they meet. Let t_flight be that time.

First step is to find the relative speed of one train compared to another. Because they are moving towards one another, their speeds should simply be added.

v_relative = 30 km/h + 50 km/h = 80km/h

If we put ourselves in the position of the conductor of the first train, we would see the second train coming towards us at 80 km/h. The initial distance is 160km, from these two values we can deduce the time t_flight using the formula d=v*t (where d is a distance, v is speed and t is time)

160 km = 80km/h * t_flight

t_flight = 160km / (80 km/h)

t_flight = 2 h

Finally we apply the same formula for the bird, knowing that it flies for 2 h and has a speed of 70 km/h.

d_flight = 70 km/h * t_flight

d_flight = 70 km/h * 2 h

d_bird = 140 km

The answer is 140 km.

The purpose of this question is to teach us to find the simplest solution for a problem.