Solve this Tough Physics question...??? (calculus involved)

Question

Three small ants are each at a vertex of an equilateral triangle of. side 100 cm. The first sets out towards the second, the second towards the third and the third towards the first with a uniform speed of 5cm/s.

During their motion each of them always heads towards its repective Target snail.

find the time after which they meet and the distance they covered during this time?

FIND THE EQUATION OF THEIR PATHS??? [Y=f(x)]

i've done the first part but am unable to find the Equation of their path someone told me that I should use Calculus in it. please find the Equation....

PLEASE HELP

Answer 1

No problem.  Let θ be any angle you like, and v the speed, with r the distance to the centre.  Then by inspection dr/dt = sqrt(3)v/2, and hence

r(t) = r_0 - sqrt(3)vt/2.

Then we can see tan(dθ) = 1/2 vdt /( r_0 - sqrt(3)vt/2 ). Taking limits as dθ->0 gives tan(dθ)->dθ, so we can integrate easily to find

θ(t) = -sqrt(3)(v^2)/4 log(r_0 - sqrt(3)vt/2).

This is all that is needed (they obviously meet at time t=2r_0/(sqrt(3)v), the singularity above).  Depending on where you place your origin, you could put this in Cartesians if you really want to (though it is indeed a spiral, so it would still have to be parametric).

With the figures you have given, they will meet 40/3s after they started, having travelled 200/3cm.

Answer 2

I'll look at it. But I do know the paths are spirals.

.

Answer 3

What does it mean: "During their motion each of them always heads towards its repective Target snail."?