Curve of Pursuit

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,

Figure 1

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 M₀(x₀, y₀) and P₀(x₀, 0) are the positions of M and P, respectively, at the initial moment and that y₀ > 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 y₀ to 0 as x increases from x₀ to

that is, M catches up with P at the point x₁ on the x-axis. In this case, the length of the curve of pursuit is equal to y₀v²/(v² – a²), and the time needed for M to catch up with P is T = y₀v/(v² – a²) (duration of pursuit). If v < a, M will not catch up with P.