My duaghter has asked me to create a puzzle for my grandson’s imminent 9th birhtday. I though it might help pass a second (or hour or week) or two for friends of all ages on this forum.
It doesn’t require any calculation, but it does need careful thought. All I’ll sya is that your first or even second thoughts may not be the right ones. This probleme does in fact illustrate Fermat’s Priciple of Least Time which in turns helps to explain why light and other waves bend (refract) when moving from one medium to another. Least Time.pdf (245.6 KB)
Choose the fastest path (A, B, C Or D) on this poll. I’ll reveal the right choice after George’s birthday next week.
I hope the five of you who tackled this problem found it interesting. I can now reveal that it was in fact a recruitment exercise for a secret internationnal think tank based in Llanddewi Brefi. Those of you who chose C will shortly be approached by men in dark suits and sunglasses - be ready.
Yes, C is the right answer (or more exactly, A, B & D are not) and here is my explanation: FermatDrowningMan.pdf (31.5 KB)
This raises the follow-on question - If you were the drowning man and saw a physicist approaching along the bank, would you think “thank goodness, I’m saved” or would you start praying?
After I gave my instinctive answer I started second-guessing myself, so tried to do it with maths… A brute-force approach with a spreadsheet (for want of a graphing calculator) showed that the rescuer should aim for a point about 8.71m along the bank before diving in (I did get more decimal places, but can’t remember them off the top of my head).
But I got stuck and bogged down trying to find an exact solution - if I did it in Cartesian coordinates (x-coordinate of the point on the bank as the unknown) I wound up with a horrible quartic; and if I called the angles between the paths & the vertical θ and φ I got as far as 5tanθ+5tanφ=10 (always) and 7sinφ=2sinθ (at the minimum time) but couldn’t then solve without winding up back with the quartic. I’m wondering if I missed a trick with trig identities or something that would have led to an exact solution… Everything I got worked with my brute-force answer substituted in, but it was… inelegant.
You have effectively derived Snell’s Law, Llongyfarchiadau
If you check out the Wiki article on Snell’s Law and scroll down to “Derivation from Fermat’s Principle” you’ll find a diagram and equations. If you assign distances, you should be able to feed them into your spreadsheet and generate a curve showing the minimum time.
Thank you for the link! And I already had the minimum time, from the spreadsheet using coordinates rather than trig (4.017s approximately) - but I still want to find the solution in surd form…
Okay; only 75 days after this problem was posed - I’ve been totally out of routine due to the COVID-19 disaster and I’m just getting back to this great Forum.
Two points:
Not being blessed with the good fortune to be a photon, I would probably choose the straight path B - wouldn’t a mere mortal actually lose time by straightening out if path C were chosen?
Where I was growing up, the man in the canal would have landed on a supermarket trolley or a mattress and would not have been able to drown.
