Solution:

Case 1: Bird flies at a speed greater than that of the train

The train (at a speed of 60 miles per hour) travels 60 miles in 60 minutes.
Therefore, the train travels from Atena to Barcena (84 miles) in 84 minutes.

Importantly, the bird makes the journeys continuously back and forth for this same amount of time (namely, 84 minutes).
Thus, the total distance traveled by the bird
= 80 miles per hour × 84 minutes = 80 × 84 / 60 miles = 112 miles.

Case 2: Bird flies at a speed less than that of the train

In 36 minutes, the bird travels 36 miles, the train travels 48 miles, and the two meet.
Now, the train (which is traveling at a speed greater than that of the bird) will reach Barcena before the bird.
So, the bird simply returns to Barcena (a return journey of 36 miles).
Thus, the total distance traveled by the bird is 72 miles.

Food for thought:

How many journeys back and forth does the bird make in Case 1?
Would the distances in these back and forth journeys form an infinite series with a finite sum?

Wish to try a simple puzzle similar in concept to Case 2? Then click here.