1) Suppose you have an ant that walks at a constant speed v on an elastic that keeps extending at a fixed speed V too. Can the ant ever reach the end of the elastic ?2) Application in everyday life: The universe is admittedly in expansion. Can you ever reach the bounds with a spacecraft ?

1) Depends on v and V. If you mean v=V, then it still depends on your reference frame. You can't just state velocities without reference. If we choose a common reference frame (the workbench say) then if the ant is moving at the same speed as the end of the band then it will never reach the end. If however v>V, for example because the ant is walking at a speed v relative to the band locally (i.e. its "leg speed") then clearly it can reach the end since it is now moving faster than V for all positions along the band.2) Firstly who says there are bounds? Main stream theories all point to there being no bound to reach. However, your "analogy" can be solved without considering the full extent of the universe. Suppose we define a large arbitrary sphere a distance D from us, and we attempt to leave that sphere. Then by our analogy above we have to travel at the speed that the boundary is moving away or faster. Make D large enough and this speed exceeds the speed of light. The hubble constant is about 70km/s/MPc and the speed of light is 300,000 km/s so we need D to be 4285Mpc or about 14,000,000,000 light years. This is the extent of the "observable universe" or hubble radius. There is no reason to believe that the universe stops there, because that would make our position in the universe special - our neighbour in andromeda galaxy would be close enough to the boundary that they could reach it (or even see it with a telescope). The result is that some parts of the universe are expanding away from each other faster than the speed of light.

QuoteOriginally posted by: iwanttobelieve1) Suppose you have an ant that walks at a constant speed v on an elastic that keeps extending at a fixed speed V too. Can the ant ever reach the end of the elastic ?2) Application in everyday life: The universe is admittedly in expansion. Can you ever reach the bounds with a spacecraft ?P.S. I love the idea that you find the notion of relativistic travel in an expansion of a not neccessarily euclidean space-time more "everyday life" than an ant and an elastic band...

I think the ant will always reach the end. And the time required for that would be something like exp(V/v).

Precisions:Ok, I add some precisions. Interview questions are always like that because the interviewer seldomly bothers giving all the asumptions (my point of view).Let's say there is a fixed end. a. The moving end is moving at speed V from this fixed end (called origine here after).b. The ant has a speed v, but she's walking on the elastic so that her speed with respect to the fixed end may be something like v + V l/L where l is her distance from origin and L is the current lenght of the elastic (this is a perfect elastic).I think computations are now easy Now, big V is bigger than little v for the application.

agree with you.(crossed my edit, with your reply)

The way to do that is to solve a simple ODE. The ant will reach the end for any (positive) v, V and the time it will take is something liket = (exp(V/v) - 1) * (L/V) where L is the initial length of the band.

Incidentally, this problem was once offered to the Russian physicist Andrei Sakharov (apparently, at some boring meeting.) Did not amuse him for too long, though. That is what he came up with in less than one minute: Edit: somehow the image is getting lost Please see attachment.

Yes, the result is that if the universe can be modelled as an expanding elastic, to which MikeCrowe do not concur, we, little ant, could reach the bounds and this is rather counter-intuitive.

QuoteOriginally posted by: iwanttobelieveYes, the result is that if the universe can be modelled as an expanding elastic, to which MikeCrowe do not concur, we, little ant, could reach the bounds and this is rather counter-intuitive.Actually if you read it I make no such comment. The point is that you can never reach the boundary of the universe because the boundary of the universe is expanding faster than the end of the elastic band. To put this in terms of your ant analogy, imagine an infinite elastic band stretching at a rate k metres per second per metre (or equivalently draw a line on it at some point and set the speed of that point). Now on the band is an ant and a beetle. The beetle starts a distance D from the ant, and moves away from the ant at speed c. Clearly the ant has no chance of catching the beetle. The beetle is the boundary of the universe.The misconception here I think is that the universe would have an "end" like the end of an elastic band, such that the "boundary" is fixed to the fabric of the universe and hence is just dragged along with it. Instead the only sensible definition of a boundary is the extent of the observable universe - i.e. that section that is causally connected to the observe. Clearly you can never reach that boundary at sub light speeds. Alternatively, if you define any point in the universe (such as a star), it must always be possible to reach it eventually.

Can someone post the complete solution?e.g. The differential EQ to solve is: whatever (I assume it's something like dx/dt = v + (V*x / (Vt + L0) )What is the solution x(t) (I have taken only one Diff. EQ course and it was many years ago, sorry I know these boards are a bunch of ex-physicists who live and breath diff EQ but I just forgot most of the techniques)

I would respectfully defer to the thread's author here...

To solve completely the problem feel free to add initial conditions such as L_0 ect... I am running a little bit short of time to post a recap answer, but I think that this topic doesn't need one recap, all seems to have been said, even if not explicitely stated. Sorry about that.

QuoteOriginally posted by: iwanttobelieve1) Suppose you have an ant that walks at a constant speed v on an elastic that keeps extending at a fixed speed V too. Can the ant ever reach the end of the elastic ?2) Application in everyday life: The universe is admittedly in expansion. Can you ever reach the bounds with a spacecraft ?The answer is yes without solving PDE. One can not find a point of elastic the ant can not reach in finite time.