Page:Elementary Text-book of Physics (Anthony, 1897).djvu/71

§ 50] If $$p_{0}$$ be the perpendicular let fall upon the line of direction of the moving particle at a chosen time when its velocity is $$v_{0}$$, and if $$p$$ and $$v$$ be the perpendicular and velocity at any other time, we know from § 48 that (35) $$vp = v_{0}p_{0}$$. If we substitute $$v = \frac{v_{0}p_{0}}{p}$$ in the above equation, we have $$\frac{v^2_{0}p^2_{0}}{2p^2} - \frac{m}{r} = C,$$ or This equation takes different forms depending upon the value of $$C$$. It becomes for

In these equations $$C$$ has now a positive value. The first equation represents an hyperbola, the second an ellipse, and the third a parabola (Puckle's Conic Sections, §§ 304, 271). The focus of each of these conic sections is the point from which $$p$$ and $$r$$ are measured, or the centre of force.

The criteria which determine the nature of the curve may be otherwise given by $$\frac{v^2}{2} > \frac{m}{r}$$ for the hyperbola, $$\frac{v^2}{2} < \frac{m}{r}$$ for the ellipse, and $$\frac{v^2}{2} = \frac{m}{r}$$ for the parabola. That is, for the three curves respectively, the velocity of the particle at a point in its path is greater than, less than, or equal to the velocity which it would acquire by falling to that point from an infinite distance under the action of the central force.

The elements of the path may be obtained from these equations. The latus rectum of the parabola is $$\frac{2v^2_{0}p^2_{0}}{m}\cdot$$ The