Quizzes & Puzzles7 mins ago
Tunnel - How Far?
The tunnel exits above ground in bright daylight. It is flat and straight
What is the maximum length that I could see the light at the end of the tunnel from the start?
I have good eyesight if that matters. There is nothing in the tunnel causing snn obstruction
Answers
No best answer has yet been selected by barry1010. Once a best answer has been selected, it will be shown here.
For more on marking an answer as the "Best Answer", please visit our FAQ.I am thinking that unlike the horizon, which is huge, the light at the end of the tunnel will be small because it is far away. When I have been through long tunnels the distant opening looks tiny when I first see it, growing larger the closer I get. Could there be a point where it is too small to register?
OK, “apparent” size, then.
As an object gets further away its apparent size decreases with the square of the distance. Assuming the tunnel appears to be 2.5 metres (roughly 8 feet) in diameter at a distance of ten metres, at a distance of 5,000 metres (roughly three miles or the distance of the "horizon") it will appear to be 2.5/(5000/10)²
2.5/(5000/10)² = 2.5/500² = 2.5/250,000
If my arithmetic is correct at this early hour I make the apparent size of the “light at the end of the tunnel” at 5km 0.00001 metres or 0.01 millimetres. That is one hundredth of a millimetre. I’ve no idea what the limits of resolution for the human eye are but I imagine it is far higher than that. A quick Google suggests it is about 0.1 mm, which seems reasonable.
So a further quick "backwards" calculation suggests the apparent size of the tunnel’s end would shrink to 0.1mm at about 1.5km (1.57km I think is near enough). So that would seem to be the distance at where the light at the end of the tunnel is no longer visible to the naked eye.
So the curvature of the Earth is not a consideration as the tunnel end becomes invisible long before the “horizon” sees it disappear.