# curve of pursuit

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## Curve of Pursuit

a plane curve that can be defined kinetically the following way. Suppose that a point *P* moves along the *x*-axis with a constant velocity *a* > 0 and that in the same plane a point *M* moves with a velocity v that is constant in absoluté value. If the velocity vector of *M* is always directed toward *P,*

the path *M* is called a curve of pursuit (Figure 1). If the coordinates of *M* are denoted by *x* and *y*, the differential equation of the curve of pursuit has the form

where *v*= ǀvǀ.

Suppose that *M*_{0}(*x*_{0}, *y*_{0}) and *P*_{0}(*x*_{0}, 0) are the positions of *M* and *P*, respectively, at the initial moment and that *y*_{0} > 0.

The equation of the curve of pursuit then has the form

when *v* ≠ *a*, and

when v = *a.* If *v* > *a, y* decreases from *y*_{0} to 0 as *x* increases from *x*_{0} to

that is, *M* catches up with *P* at the point *x*_{1} on the *x*-axis. In this case, the length of the curve of pursuit is equal to *y*_{0}v^{2}/(*v*^{2} – *a*^{2}), and the time needed for *M* to catch up with *P* is *T* = *y*_{0}*v*/(*v*^{2} – *a*^{2}) (duration of pursuit). If *v* < *a, M* will not catch up with *P.*