Page:Mars - Lowell.djvu/283

APPENDIX

Note I

The critical velocity at the surfaces of the planets is found as follows:—

Using the usual symbols we have:

$fdt=dv\,$

$\therefore fds=vdv$.

And as $f={\frac {-m}{s^{2}}}$, since the force tends to decrease the coördinates, this becomes $-{\frac {mds}{s^{2}}}=vdv$
Integrating:

${\frac {m}{s}}={\frac {1}{2}}v^{2}+c$, of which the definite integral from s₁ to s₂ is

${\frac {m}{s_{1}}}-{\frac {m}{s_{2}}}={\frac {1}{2}}v_{1}^{2}-{\frac {1}{2}}v_{2}^{2}$

Hence, since at infinity the velocity is 0, the equation for a fall to a planet's surface from infinity is

${\frac {m}{r}}={\frac {1}{2}}v^{2}$,

r being the radius of the planet and v the velocity acquired at its surface from a fall from infinity, which is the same as the velocity needed for projection from its surface to infinity.

To find m we have in the case of the Earth g = 32 ft. a second at its surface; this gives us m in terms of g, that is, f. For the other planets we need only to introduce their masses and radii in terms of those of the Earth and then multiply the value for the Earth by the square root of the ratio.

The result is that we find the critical velocity for the several planets and for the Sun to be as follows:—

Mercury	2.2	miles	a	second	(probable	value).
Venus	6.6	"	"	"	"	"
Earth	6.9	"	"	"
Moon	1.5	"	"	"