Solving a physics (kinematics) problem using series

Question

I want to solve the following problem:

Two trains move in a straight line and opposite direction at 30 km/h from two points 180 km away. A bird comes from one of these points at 60 km/h heading to the train coming in the opposite direction. When the bird reaches the train, turns around and heads to the other train, repeating the process indefinitely. How many kilometers will the bird have flown until the fatal outcome? How many in each direction?

I am supposed to use series in this problem, but I don't know how to start. The only thing I know is I should use motion equations for the trains and the bird.

I would be most grateful if you help me. Thank you in advance.

Answer 1

Hints:

• How far does the bird fly between starting and the first turning around and reversing direction of flight?
• How far does the bird fly between the first and second turns?
• How far does the bird fly between the $n^\text{th}$ and $(n+1)^\text{th}$ turns? Is there a constant ratio between the durations of successive flights between turns?

A picture might help. (In fact, drawing a picture frequently helps us organize our thoughts and verify we understand the data presented in a problem.)

One train (orange) starts at position $0 \,\mathrm{mi}$ and its position increases by $30 \,\mathrm{mi}$ each subsequent hour. The other train (green) starts at position $180 \,\mathrm{mi}$ and its position decreases by $30 \,\mathrm{mi}$ each subsequent hour. The bird (blue) flies between the trains alternately increasing or decreasing its position by $60 \,\mathrm{mi/h}$

Final hint:

• For the "in each direction" question, is there a common ratio between the duration of flight in one direction and the duration of the next flight in the same direction (i.e., between the $n^\text{th}$ and $(n+2)^\text{th}$ flight segments)?

Answer 2

The bird approaches one train at relative speed 90 km/h, reaching it in 2 hours after flying 120 km. The trains are only $\frac13$ as far apart now, because they'll collide in an hour. This sets up a geometric series, of sum $\frac{120\,\text{km}}{1-1/3}=180\,\text{km}$. But it's easier to notice the bird will fly for three hours.