You're blindfolded, disoriented, and standing one mile from the Great Wall of China. How far do you have to walk to guarantee that you will run into the wall?
Assume that the wall is infinitely long and straight.
(In reply to Apparently not the shorter solution. Sorry.
I think your physics professor is getting very upset. This solution is not even as good as your last one.
If the wall runs east west, and you are a mile south, and you start walking east, this solution never gets you closer than 1 mile from the wall.
In order for this approach to work, the fourth leg must by 2.83 miles and then there must be a 5th leg (one more right turn) that is somewhere between 1.414 and 2.83 miles.
Of course, you can cut corners, since once you start turning you never need to stray further than 1 mile from your starting point.
I believe that Harry's solution is the winner. Similar to Harry's solution, if you only want to make 90 degree turns, then there may be some improvement available by going more than 1.414 miles on the first leg, which allows the 5th leg to be shorter.