Point A moves uniformly with velocity `nu` so that the vector `v` is continually "aimed" at point B which in its turn moves rectilinearly and uniformly with velocity `upsilon lt nu`. At the initial moment of time `v_|_u` and the points are separated by a distance `l`. How soon will the points converge?

Answer

It is seen from figure that the points A and B converge with velocity `v-u cos alpha`, where the angle `alpha` varies with time. The
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points merge provided the following two conditions are met:
`underset0oversettauint(v-ucosalpha)dt=l`, `underset0oversettauint v cos alpha dt=utau`,
where `tau` is the sought time. It follows from these two equations that.
`tau=vl//(v^2-u^2)`.

Explanation

Let us locate the points A and B at an arbitrary instant of time (figure).
If A and B are separated by the distance s at this moment, then the points converge or point A approaches B with velocity `(-ds)/(dt)=nu-upsilon cos alpha` where angle `alpha` varies with time.
On integrating,
`-undersetloverset0intds=underset0oversetTint(nu-upsilon cos alpha)dt`,
(where T is the sought time.)
or `l=underset0oversetTint(nu-upsilon cos alpha)dt`
As both A and B cover the same distance in x-direction during the sought time interval, so the other condition which is required, can be obtained by the equation
`Deltax=int v_xdt`
So, `upsilonT=underset0oversetTintvcosalphadt` (2)
Solving (1) and (2), we get `T=(upsilonl)/(nu^2-upsilon^2)`
One can see that if `upsilon=nu`, or `upsilonltnu`, point A cannot catch B.