(18)
no additive constant being necessary if t be reckoned from the instant of starting, when φ = 0. The time t of reaching the origin (φ = π) is
| t1 = | π c3/2 | . |
| √(8μ) |
(19)
This may be compared with the period of revolution in a circular orbit of radius c about the same centre of force, viz. 2πc3/2 / √μ (§ 14). We learn that if the orbital motion of a planet, or a satellite, were arrested, the body would fall into the sun, or into its primary, in the fraction 0.1768 of its actual periodic time. Thus the moon would reach the earth in about five days. It may be noticed that if the scales of x and t be properly adjusted, the curve of positions in the present problem is the portion of a cycloid extending from a vertex to a cusp.
In any case of rectilinear motion, if we integrate both sides of the equation
| mu | du | = X, |
| dx |
(20)
which is equivalent to (1), with respect to x between the limits x0, x1, we obtain
1⁄2 mu12 − 1⁄2 mu02 = ∫x1x0 X dx.