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Chasing Particles

Brilliant dwellers, can anyone help me with a physics question?

In a 2D plane, a particle moves along the \(x\) axis at the speed of \(1 \, m/s\), with its initial position at \((0, 0)\). Another particle, with initial position \((0, 1)\), starts to chase the first particle with a constant speed of \(2 \, m/s\) in such a way that its trajectory is defined instantaneously by the vector created by the particles (all of the units of the plane are in meters). Ignoring any interactions between the particles, as well as forces such as gravity, answer the following:

1) What kind of curve does the chasing particle define in the plane?

2) At which point do the particles meet?

3) How long does it take until the chasing particle catches up to its target?

I really have no clue as to how to work this out; I tried applying Calculus concepts, but I couldn't come up with a solution. If anyone can help me out here, I'd appreciate it a lot.

Note by Alexandre Miquilino
2 years, 5 months ago

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2) at the point (1/sqrt(3),0) the two particles will meet. 3) The time t=1/sqrt(3). Here, you have to take relative velocities between the particles and work using those equations. Mahathir Ahmad · 2 years, 5 months ago

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@Mahathir Ahmad Could you be a little bit more clear as to how you've achieved those numbers, please? Alexandre Miquilino · 2 years, 5 months ago

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@Alexandre Miquilino The name of the curve you are looking it is the so called tractrix.

I do not want to spoil the pleasure of solving the problem, the one you propose, is dated since Leonardo da Vinci times.

I am sure that a search under pursuit curve or tractix you will get information in the web.

In times when the web was not available I would recommend you the book Harold T. Davis Introduction to non linear Differential and Integral Equations from Dover. ( pages 113 thru 127) contains a full detailed treatment of your case and other cases of pursuit. Come back is you are unsuccessful. Mariano PerezdelaCruz · 2 years, 5 months ago

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Hey! This problem is actually pretty similar to one I posted a while ago, so here's the link to that. I posted a solution too, so you could see if that helps you out. Cheers! Raj Magesh · 2 years, 5 months ago

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@Raj Magesh Raj, thank you so much for the solution! The explanation is very clear and concise, and was what I needed exactly. I can't really thank you enough for the help, I do appreciate it a lot. Alexandre Miquilino · 2 years, 5 months ago

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@Alexandre Miquilino No problem, Alexandre, I'm glad to help! :D There is a similar problem in I.E. Irodov's Problems in General Physics. After trying it for a looong, loooong time (and failing), I looked at the solution and was amazed by the relative velocity equation, which I wouldn't have thought of myself in a million years... Raj Magesh · 2 years, 5 months ago

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@Raj Magesh Neither would I! I tried solving some bizarre differential equations, but I couldn't get them to work at all... this method is "Brilliant" haha Alexandre Miquilino · 2 years, 5 months ago

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