# Classical mechanics / system of particles

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?

• 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.
– Dan
Commented Dec 20, 2021 at 0:14

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.