A geometry puzzle goes like this: at time $t=0$ point $A$ is at the origin $(0,0)$ and point $B$ is at $(0,1)$. At time $t=1$, $A$ starts to move to the right with velocity $1$, and $B$ starts to move toward $A$ with the same velocity. What will be the distance between them when $t$ goes to infinity?
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Perhaps it is solvable by differential equation, but that's not the point. There are at least two interesting solutions that doesn't involve differential equation, although they're arguably the same solution.
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My solution:
We describe everything from $A$'s perspective.
A stays at $(0,0)$ forever, while $B$ starts from $(0,1)$ and has velocity vector $(-1,0)+\alpha(-x,-y)$ when it is at $(x,y)$. $\alpha$ is a positive number such that the second terms has length $1$, i.e. $B$'s velocity vector bisects vectors $(-1,0)$ and $(-x,-y)$.
Recall parabola can be defined as a curve consisting of points with equal distance to a fixed point, focus, and a line, directrix. Moreover the tangent line of any point on parabola bisects the rays from the point to the focus and directrix.
So, taking initial condition into account, $B$ is on the parabola with focus $(0,0)$ and directrix $x+1=0$. It moves downward and leftward and approaches $(-0.5,0)$ as time goes to infinity.
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An even more interesting solution, not by me, requires no knowledge of parabola:
By analyzing $B$'s velocity vector also from $A$'s perspective, the amount that $B$ has moved to the left equals what it has moved toward $A$. So at $t=\infty$, suppose it's at $(-k,0)$ with $k>0$. Then $k=1-k$, i.e. $k=0.5$.
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An even more interesting solution, not by me, requires no knowledge of parabola:
By analyzing $B$'s velocity vector also from $A$'s perspective, the amount that $B$ has moved to the left equals what it has moved toward $A$. So at $t=\infty$, suppose it's at $(-k,0)$ with $k>0$. Then $k=1-k$, i.e. $k=0.5$.
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