Three insects A,B and C are situated at the vertices of an equillateral triangle of side 1. the insect A heads towards B,B towards C,C towards A with constant speeds v such that they always remain at the vertices of an equilateral triangle. Find the (a) time of their meeting (b) equation of path traced by one insect relative to the other.

Explanation

After a time t, let the distance of spearation between the insects be r. the velocity of approach (component of relative velocity `u_(rel)=v+v cos 60^(@)=(3v)/(2)`

Since, `(v_(rel))_(||)dt=-dr`, submitting `(v_(Rel))_(||)=(3v)/(2)`, we obtain
`(3v)/(2)dt=-dr` ......(i)
As ,` r=l "at" t=0` and `r=0 "at" t=tau` (time of meeting ), integrating both sides, we have
`(3v)/(2)int_(0)^(tau)dt=-int_(l)^(0)dr`
This gives, `tau=(2l)/(3v)`