### Brain Teasers

# Crossing Paths

A person starts off walking in a perfectly straight line at uniform speed. After a mile, a second person starts walking at uniform speed from the same starting point and catches up with the first guy after another mile. At this time, the second person turns at an angle of 60 degrees west. They both continue walking in straight lines for a certain period of time. Then, the second person turns 60 degrees east. They both continue in straight lines. Will the second person cross the first's path in front, behind or at the same time?

### Answer

They will intercept!Person 2 catches up with the first guy after another mile

This tells us that person 2 is walking twice the speed of 1.

He goes off at an angle of 60 degrees, but his speed in the same direction as the first person is 2cos60

cos60 is 1/2 so the speed in the same direction is the same.

When he turns, his speed is 2sin60 = 1!

It doesn't matter when he turns, the paths will eventually meet up again!

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