Any math whizzs here? Two bugs are on an east-west line a distance d apart. both travel at the same speed.---

Question

R travels due north on the y axis. S travels in a direction directly towards R at all times. S eventually approaches the y axis going
(almost) due north. At that time, how far ahead is R?
I graphed this and came up with .41d.
Can anyone tell me what function or series represents S's course, and how they solved this?
Thanks for any help you can give me.

Answer 1

It looks to me like it's .5d which is not too far from your answer.

For simplicity, use a unit of length in which d = 1 and a unit of time in which the speed of each of the two bugs is also 1. Let r be the distance between them and let h be the angle between the y direction and the vector from S to R. In other words h starts out pi/2 and approaches 0.

The time derivatives are:

dh/dt = -sin(h)/r , dr/dt = -1 + cos(h)
--->
dr/dh = r(1-cos(h))/sin(h)

Separating the variables r and h to opposite sides of the equation and integrating both sides with the initial condition (r=1 at h=PI/2) gives:

r = tan(h/2)/sin(h)

You want the distance between the two bugs as t-->oo or equivalently as h --> 0 which is r=1/2 (use L'hopital's rule for example), or d/2 in terms of the initial distance d.

Of course S never actually reaches the y axis but approaches it asymptotically.

(Maybe I should add, I first tried to formulate the problem in terms of the Cartesian coordinates and was able to write a differential equation for the curve but it looked very difficult to solve. Finally tried using the above coordinates (r,h) and it became much easier)

Answer 2

you purchased me stuck in this math issue so i am going to provide it a shot. in view that A and B are happening an similar % i trust that Bs route might want to be a million/4 of a circle. because the circumference of a circle (C) is C=2x3.14xd or 6.28d, then a million/4 of which could be a million.57d. once B traveled that quarter of a circle it would want to be good in the back of A going North. in view that B traveled a million.57d so did A; besides the undeniable fact that, B has purely lengthy gone the radius of the circle (d) North. hence the version between both is: a million.57d - 1d = .57d it really is my best shot. solid success.