Can a Spaceship Always Spot the Lone Inhabitant on a Spherical Planet?

[Heirot]

Homework Statement

Spherical planet has only one inhabitant that can move freely on the surface of the planet with speed u. A spaceship approaches the planet with velocity v. Show that if v/u > 10, the spaceship can always see the inhabitant regardless of his movement.

Homework Equations

No idea.

The Attempt at a Solution

I have no idea where to start. Is this supposed to be a relativistic problem or a classical mechanical one? Any hints are more than welcome!

 

[BishopUser]

What class/chapter is this for? Has me stumped atm unless the problem statement is incomplete.

 

[Heirot]

Our professor gives us all kinds of problems so that we become versatile physicists like Landau. This one came under mathematical physics.

 

[Borek]

Problem is incomplete. That suggests you should make some assumptions on your own. However, it is pretty easy to construct initial conditions such that the ship doesn't see the inhabitant the moment it starts, so obviously you can't prove the statement. OTOH, if ship is close enough (it hovers just above the inhabitant head) it is enough that v/u > 1.

 

[Heirot]

Sorry, I wasn't clear enough - the ship should see inhabinant eventualy, not neccessarily at the initial moment. If he's on a planet, he can't hide forever. With that said, I presume the worst initial conditions result in a factor 10.

 

[diazona]

Ah, well that's not what the problem said originally (it said "always see the inhabitant").

Anyway, to provide a really solid solution, I think you'd have to start with a little game theory, to figure out the optimal escape strategy for the planet's inhabitant and the optimal search strategy for the ship to counter it. Already that seems way outside the scope of a physics class. (Interesting problem, though)

I suppose you could think about how much of the planet can be seen at a time from the ship, and see if that leads to anything... although I thought about it a bit and I still don't see where the 10 comes from.

 

[Heirot]

I think that 10 is an approximation of pi^2.

One of my guesses would be that if the planet has a radius R and the ship is displaced by d from the center of the planet, then, one can calculate the amount of the planet surface seen from the ship (let's call it ship's area). It decreases with d. The worst case scenario is that the inhabitant is initially on the other side of the planet. If the ship can get to the other side faster then the inhabitant can escape the ship's area located in the point where the inhabitant was at the beginning then the ship can locate the inhabitant. Unfortunately, this doesn't give pi^2 nor 10.

 

[diazona]

Hm, π² at least seems less random...

One of my guesses would be that if the planet has a radius R and the ship is displaced by d from the center of the planet, then, one can calculate the amount of the planet surface seen from the ship (let's call it ship's area). It decreases with d.

Actually it increases with d, doesn't it? Take Earth as an example: if you're just above the surface, you can only see a few miles in any direction at best, but from the ISS in low orbit, you can see entire continents. And looking from the moon, you see nearly half the Earth's surface. So the further away you are, the more you see.

Of course, I still can't think of anything that would really clarify this problem either...

 

[Heirot]

Yes, you're right - I mistyped.

 

[Borek]

Hm, π² at least seems less random...

Hm, so I was not the only person to think that 10 looks ridiculously round for an answer

 

[diazona]

Yeah, it's awfully suspicious-looking, sitting there nonchalantly without a decimal point