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My friend is doing a math project on planes and missiles, and I posed two interesting questions. Here they are, more formally phrased.

A plane and a missile are in the coordinate plane. At time $t=0$, the plane starts at $(0,0)$ and moves at one unit per second to the right. Also at $t=0$, the missile starts at $(0,1)$ and travels at one unit per second towards the plane's current location at all times.

  • What is the resulting path of the missile?
  • After a long time, the missile's path approaches the line $y=0$. However, the missile will always be a bit behind the plane. As $t\to\infty$, how much does the missile trail the plane by? (In other words, what is $\lim_{t\to\infty}\text{dist}(\text{plane},\text{missile})$?)

We conjectured that the path was exponential, but the calculus proved to be far too challenging. We found that $\frac{dx}{dt}=\frac{t-x}{\sqrt{(t-x)^2+y^2}}$ and $\frac{dy}{dt}=\frac{y}{\sqrt{(t-x)^2+y^2}}$, but we weren't able to make any progress from there.

I made a Scratch project to test this, and it looks like the answer to the second problem is $\frac 12$, though that's just speculation.

Can anyone make any further progress on this problem? Thanks!

ajxu2
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  • Look up "curves of pursuit" – Paul Nov 07 '23 at 08:11
  • An interesting question is if the plane moves on a closed path (periodic motion) what is the limiting path of the missile? If the plane moves on a circle it is fairly easy to see that the missile limiting path is a circle. – Paul Nov 07 '23 at 10:31

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It seems like this answer from an old post should point you to everything you need Pursuit curves solution . The problem is almost exactly the same except it considers more general cases (non-equal speeds of chaser and chasee) etc and also is rotated.

For your second question, since you can expect the distance to be the same in the rotated problem, you can use the expression of the curve from the linked post in the case of equal speed and $p=1$, as well as the expression for $t$ in terms of $(x, y(x))$ (which is the curve followed by the chaser) to express

$$\text{distance between chaser and chased} = \| (x, y) - (1,t) \| = \\ \|(x, y)-(1, y+(1-x)\frac{dy}{dx}\| \\ = \sqrt{(x-1)^2 + (x - \frac{1}{2}x^2)^2} \rightarrow \frac{1}{2} \text{ as }x \rightarrow 1$$

as your Scratch project also suggests. I've never seen these 'pursuit curves' before - interesting dynamics!

brighton
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