×

# Hi Brilliantians!

Today I have developed a general formula for the classic Irodov problem-

3 particles are kept at the vertices of an equilateral triangle of side $$a$$.The velocity vector of the first particle continually points towards the second,the second towards the third and the third towards the first.After what time will they meet.Each moves with a constant speed $$v$$.

Well for a regular $$n$$ sided polygon ,

# $$T =$$ $$\frac{a}{v_{0}(1-cos\frac{2\pi}{n})}$$

Working :

Well the usual stuff,figuring out relative velocities and then dividing the initial distance by the $$v_{rel}$$ to get the time.Becuase the trajectory forms logarithmic spirals and maintains its symmetricity all throughout the motion.

Note that each angle of the polygon is given by $$\frac{(n-2)\pi}{n}$$ therefore the exterior angle will be $$\frac{2\pi}{n}$$.Resolving the velocity component along this direction,we get

$$v_{rel} =$$ $$v - v cos$$ $$\theta$$ where $$\theta$$ $$=$$ $$\frac{2\pi}{n}$$

hence $$t =$$ $$\frac{a}{v_{rel}}$$ giving us our desired expression.

Note that for ,

$$n = 3$$ we have $$t = 2a/3v$$.

$$n = 4$$ we have $$t = a/v$$.

$$n = 6$$ we have $$t = 2a/v$$

$$n = 12$$ we have $$t =$$ $$\frac{2a}{(2-\sqrt 3)v}$$

$$n = infinity$$ we have $$t =$$ $$infinity$$ because now the path is a circle.

and so on.

# Thanks everyone for reading!

Note by Ayon Ghosh
3 weeks, 4 days ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$