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Three particle A, B and C situated at the vertices of an equilateral triangle starts moving simultaneously at a constant speed $v$ in the direction of adjacent particle, which falls ahead in the anti-clockwise direction. If $a$ be the side of the triangle, then find the time when they meet.

I know how to solve this problem using the symmetry and other concepts. But how can I construct the equation of motion for a particle without using trivial symmetry but deriving it analytically like we do in a two body problem. Without using any symmetry or relative velocity concept, how can we generate the vector differential equation for a particle?

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  • $\begingroup$ Maybe a rotating reference frame will help? Suppose you use a frame where the particles all maintan the same orientation to the coordinate axis. In such a frame you should get an effective motion directly in or out. $\endgroup$
    – Dan
    Commented Dec 20, 2021 at 0:14

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Your equilateral triangle will rotate and shrink. The mass at each vertex will have a constant radial speed ${v_r} = – (v)cos(30^o$) and a tangential speed ${v_t } = (v)sin(30^o$). The radius for each (measured from the center of the triangle) will be r = ${r_o} – {v_r}t$, where ${r_o}cos(30^o$) = a/2. For the angular velocity ω = ${v_t}$/r giving dθ = ${v_t}(dt)/({r_o} – {v_r}t)$. Integrate both sides of this equation (with appropriate limits) to get θ as a function of time. With r and θ as functions of time, you can also get x and y as functions of time.

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