Undergrad Can a Cat Catch a Mouse Running in a Circle at the Same Speed?

[BillhB]

So the question goes something like this:

Suppose a mouse runs in a circle of radius R, with some constant speed $||V_m||$. A cat chases the mouse starting at the center of the circle and also moves at a constant speed $||V_c||$, such that $||V_m||=||V_c||$, so that it is always between the center of the circle and the mouse. How long will it take for the cat to reach the running mouse?

Now I've already found the solution to the problem, so I don't need any assistance there, and why I'm not posting this in homework help. What I'm having trouble with is visualizing the situation at some instant right before the cat catches the mouse.

It seems to me that if I look at some $d\theta$, when the mouse and cat are some $dx$ distance apart, they would need to run parallel to one another at the same velocity to maintain the co-linear requirement (basically the limiting case where the distance approaches zero?). So it would seem like the cat and mouse would never actually meet. I'm thinking this since if it stepped off the current track it would need to violate one of the two conditions, or end up some $dx$ behind the mouse. I haven't been able to reconcile the above to my satisfaction.

My first thought was maybe it's because the curves are only approximately straight lines at some small angle, and the components of the velocities are only approximately equal at some $dx$, such that the cat would still have a small bit of velocity left over to approach the mouse and still remain co-linear with the center of the circle until they actually meet- as the cats radius would still be some $dx$ smaller than the mouses. I can't think of a good way to see this though.

Is there some concept I'm missing here? Does this even make sense to anyone else?

 

[drvrm]

My first thought was maybe it's because the curves are only approximately straight lines at some small angle, and the components of the velocities are only approximately equal at some dxdxdx, such that the cat would still have a small bit of velocity left over to approach the mouse and still remain co-linear with the center of the circle until they actually meet- as the cats radius would still be some dxdxdx smaller than the mouses. I can't think of a good way to see this though.

suppose the event is happening on earth
will the cat can catch the mouse?
moreover you are only restricting the modulus/magnitude of velocities not its direction - so any direction can be taken!
the third thing is what is the initial condition of the chase?

 

[BillhB]

suppose the event is happening on earth
will the cat can catch the mouse?

Physically the problem is dumb, I realize that. I'm looking at the abstraction though, more specifically, the last instance of chase. So we could even replace the cat and mouse with particle A and particle B if we really wanted to.

moreover you are only restricting the modulus/magnitude of velocities not its direction - so any direction can be taken the third thing is what is the initial condition of the chase?

For the first part, I don't see a problem with that, if you restrict the direction of velocity for the cat the problem becomes truly unsolvable. The mouse is running in a uniform circular motion path, so whether it starts running to the left or right at time = 0 also seems irrelevant.

The conditions are above, the mouse and cat have the same speed, the mouse is running some fixed circle with radius R, the cat wants to catch the mouse and also wants to remain co-linear with the mouse and the center of the circle.

 

[Svein]

My first thought was maybe it's because the curves are only approximately straight lines at some small angle, and the components of the velocities are only approximately equal at some dxdx, such that the cat would still have a small bit of velocity left over to approach the mouse and still remain co-linear with the center of the circle until they actually meet- as the cats radius would still be some dxdx smaller than the mouses. I can't think of a good way to see this though.

Are you re-inventing Zeno's paradox?

 

[BillhB]

Are you re-inventing Zeno's paradox?

I'm trying not to, I saw the similarity as well...

I just wanted to know how can we avoid the paradox?

 

[drvrm]

I just wanted to know how can we avoid the paradox?

because there are an infinite number of points cat must reach where the rat has already been, the cat can never overtake the rat!
how you wish to solve it-your move!

 

[BillhB]

because there are an infinite number of points cat must reach where the rat has already been, the cat can never overtake the rat!
how you wish to solve it-your move!

That's not what the problem is. The problem is that the cat can never cross the last $dx$ of distance without violating the co-linear or velocity conditions to catch the mouse. It has no intention of overtaking it.

There must be some non hand-wavy explanation that reconciles the problem in a somewhat rigorous way.

 

[micromass]

According to the hyperreal theory of infinitesimals, an infinitesimal time before the cat catches mouse, the angles willbe parallel except for an infinitesimal amount.

 

[BillhB]

According to the hyperreal theory of infinitesimals, an infinitesimal time before the cat catches mouse, the angles willbe parallel except for an infinitesimal amount.

Hey, thanks for the reply.

So I looked at hyperreals, and the transfer principle seems to say that all the axioms of the reals extend to the hyperreal. So it would seem that right before the cat catches the mouse, if the angles were parallel except for some infinitesimal amount then $\theta_m=\theta_c + d\theta$, but since $x+0=x$ transfers over from the reals, then $\theta_c+d\theta = \theta_c$ and then $\theta_c=\theta_m$. So I feel like I'm in the same boat, or I could be butchering this.

I feel like this may be a case of me needing to let this go, as I probably don't yet have all the tools to resolve the problem very well. I'm only in calculus three and haven't studied much mathematics. It just bothered me when I thought about it.

 

[micromass]

Hey, thanks for the reply.

So I looked at hyperreals, and the transfer principle seems to say that all the axioms of the reals extend to the hyperreal. So it would seem that right before the cat catches the mouse, if the angles were parallel except for some infinitesimal amount then $\theta_m=\theta_c + d\theta$, but since $x+0=x$ transfers over from the reals, then $\theta_c+d\theta = \theta_c$ and then $\theta_c=\theta_m$. So I feel like I'm in the same boat, or I could be butchering this.

I feel like this may be a case of me needing to let this go, as I probably don't yet have all the tools to resolve the problem very well. I'm only in calculus three and haven't studied much mathematics. It just bothered me when I thought about it.

$\theta_c +d\theta = \theta_c$ is false.

 

[BillhB]

$\theta_c +d\theta = \theta_c$ is false.

Then that helps clear it up a little bit, thanks again. I'll have to do more reading.

 

[pbuk]

The limit of that "small bit of velocity left over" required is 0. This is no different to Xeno.

 

[ebos]

Maybe I'm reading it wrong, but I was wondering that if you had a smart cat it would be able to figure out where the mouse would be in the time it took him to run straight out from the center of the circle to the circumference circle. Then he could just run for 1 r and meet the mouse after it has run for 0.5 rads. Similar to sending a rocket to Mars.

 

[jbriggs444]

Maybe I'm reading it wrong, but I was wondering that if you had a smart cat it would be able to figure out where the mouse would be in the time it took him to run straight out from the center of the circle to the circumference circle. Then he could just run for 1 r and meet the mouse after it has run for 0.5 rads. Similar to sending a rocket to Mars.

The interesting bit about this problem that is being argued about implicitly assumes a smart cat and an evasive mouse.

If the mouse has a "lead" on the cat, he will exploit that lead to try to stay ahead. The cat avoids this possibility by maintaining a position between the mouse and the center of the circle. Given this, it does not matter whether the mouse ever turns around. The cat will maintain his position and the intercept time is unaffected.

If the cat were to use the "straight line chase" strategy, the mouse would turn tail and run the other way.