Chase on an endless road, continued

[Rainman]

This is a sequel to the puzzle:

You have just managed to catch up to your friend in a race, on a road which goes on forever in both directions. Your friend is a bad loser, and full of adrenaline from the race. "Try and catch me this time", your friend says, and knocks you out cold. When you recover, you have no idea how much time has passed.

• You move at constant speed x>1 m/s and your friend moves at constant speed 1 m/s.
• Visibility is limited so you will not be able to see your friend from a distance.
• You must not stray from the road, same goes for your friend.

Can you catch up to your friend? Does the answer depend on your speed x, and if so, what is the minimum x for which you can catch up? What is your strategy?

[gavinksong]

You just run down the path in one direction for a time.

If you haven't caught up to him, you turn and run back to your starting point, and then some more to compensate for the distance you may have lost, and then some more to see if you can catch up to him.
If you still haven't caught up to him, you turn and run back to where you left off in the opposite direction, compensate for lost distance, and then run for a little bit longer.
You repeat this until you've caught up to him.

Demo
You run down the path in one direction for T seconds.
If you haven't caught up to your friend, you turn and run in the other direction for T seconds to return to the starting point, and then for 2T/(x-1) more seconds to compensate for lost distance, and then for T more seconds to see if you can catch up to him.
If you still haven't caught up to your friend, you turn and run again for 2T+2T/(x-1) seconds back to where you left off, and then for 2(2T+2T(x-1))/(x-1) seconds to compensate for lost distance, and then T more seconds to see if you can catch up to him.

And then repeat, turning and running for T(n) seconds if you have turned for the nth time.

T(n+1) = T(n)*(x+1)/(x-1) + T(0)

Edited July 30, 2013 by gavinksong

[Rainman]

Caught again, your friend concedes that you are the supreme runner. Maybe it's not a coincidence that your profile pic is an ostrich

[bonanova]

I love this.

I didn't do the series to quantify things, so, being lazy I'll just ask:

Do all the questions in the OP have answers?

e.g. if x = 1.0000001, is it impossible? Or just take much longer?

[gavinksong]

I love this.

I didn't do the series to quantify things, so, being lazy I'll just ask:

Do all the questions in the OP have answers?

e.g. if x = 1.0000001, is it impossible? Or just take much longer?

It just takes longer.