Spacecraft vs Alien on a spherical planet

[magic_castle32]

Here's a problem which I encountered a while ago:

"There is an alien on the surface of a spherical planet. The alien can run at a top speed of u. An alien hunter is hunting the alien in his spaceship which can fly at a top speed of v. Once the hunter sees the alien, he fires, and the alien dies. Show that the alien will always die if v>10u."

The caveat is that the hunter does not know the position of the alien, and when the alien appears in the field of vision of the spacecaft it's gameover.

Any suggestions? The height of the spacecraft above the planet is likely to play a part; perhaps the solution is along the lines of confining the area of movement of the alien to a decreasing part of the sphere.

(not a homework question)

 

[uart]

Here's a problem which I encountered a while ago:

"There is an alien on the surface of a spherical planet. The alien can run at a top speed of u. An alien hunter is hunting the alien in his spaceship which can fly at a top speed of v. Once the hunter sees the alien, he fires, and the alien dies. Show that the alien will always die if v>10u."

The caveat is that the hunter does not know the position of the alien, and when the alien appears in the field of vision of the spacecaft it's gameover.

Any suggestions? The height of the spacecraft above the planet is likely to play a part; perhaps the solution is along the lines of confining the area of movement of the alien to a decreasing part of the sphere.

(not a homework question)

Sorry but that doesn't seem to make sense to me. Why would the alien not always die even if the hunters speed was much lower, say only 2u for example. Surely it would just take the hunter longer to find the alien. I also think it's an underspecified problem, the range at which the alien can detect the hunter versus the range at which the hunter can detect the alien should also be relevant.

 

[LukeD]

I'm a little confused about the definition of the problem. It seems to be asking to show that two random paths on a sphere traveled at speed u and v will intersect if v > 10u. However, that is definitely not true. (Consider just two non-intersecting closed paths).

We need to know something about the paths that the two take. Should we maybe assume that the hunter takes a path that covers the entire sphere? Also, does the hunter have any field of vision or does he have to physically bump into the alien to see him? If so, then we can consider a sphere in Q^2 instead of R^2. (This may be unnecessary though)

Do you have any more information for this problem?

 

[Grumm]

I have a feeling the key element here (which was unstated) would be the curvature of the planet.

The spacecraft needs to catch up to the alien, i.e. gain line of sight.

However, I don't see why any comparison of velocities would make any difference (aside from the alien merely being a fraction faster than the spacecraft )

 

[magic_castle32]

Sorry but that doesn't seem to make sense to me. Why would the alien not always die even if the hunters speed was much lower, say only 2u for example. Surely it would just take the hunter longer to find the alien. I also think it's an underspecified problem, the range at which the alien can detect the hunter versus the range at which the hunter can detect the alien should also be relevant.

I don't think v>10u is a lower bound for v - so that gives us some freedom in choosing the parameters for the problem, perhaps (say, the height of the spacecraft above the planet).

Sorry, the question received was phrased in this ambiguous manner =/

I believe the problem is to find a strategy for the spacecraft to hunt down the alien - a strategy which works so long as as v>10u. And that the alien is immediately shot down when he appears in the field of vision of the spacecraft (which, in turn, depends on the height the spacecraft is hovering over the sphere).

 

[Werg22]

That's a physical problem...

 

[LukeD]

That's a physical problem...

Nah. If you ignore the fact that the problem talks about a hunter and an alien, you can treat it as a purely mathematical problem

You can treat the path of the alien as some path on the sphere where ds/dt <= u (where s is the arc length of the path), and then show that there is some path with ds/dt <= v that intersects every path that the alien takes (at the same time t as that in the path of the alien)

I would help come up with a solution, but I have not yet studied manifolds and with the tools that I do have, it would take a while to come up with a proof.

I think the fact that the hunter has a field of vision is important though. We can then treat the planet as a sphere in Q^2 instead of R^2. Otherwise, it seems intuitive that, if the alien knew where the hunter was at all times, then no matter how slow the alien was moving, he could always move out of the way fast enough to just barely miss the hunter.

 

[magic_castle32]

I'm thinking if the spacecraft were to travel in a helix-like motion - that might do the trick. The alien would then be confined to a decreasing area of the spherical planet, and would not be able to cross over to the other side without being seen and shot down by the spacecraft if v > 10u.

Now we need a mathematical justification...