Page:Elementary Text-book of Physics (Anthony, 1897).djvu/73

§ 52] with the acceleration $$\frac{\mu + m}{r^2}\cdot$$ The path of the particle $$\mu$$ relative to the particle $$m$$ will be therefore that due to a central force proceeding from $$m$$ and equal to $$\frac{\mu (\mu + m)}{r^2}\cdot$$ The radius vector drawn to $$\mu$$ from $$m$$ will still sweep out equal areas in equal times, and the path of $$\mu$$ will still be a conic section. If its path be an ellipse, the periodic time will be given by $$T = \frac{2 \pi a^{\tfrac{3}{2}}}{(\mu + m)^{\tfrac{3}{2}}};$$ so that $$\mu + m = \frac{4 \pi^2 a^3}{T^2}\cdot$$

52. Motion of Projectiles.— In the special case of central motion in which the distance of the centre from the moving particle is very great and the velocity of the particle small, the particle describes a portion of an ellipse which differs very little from a parabola. This may be seen at once from the equation $$\frac{v^2}{2} = \frac{m}{r} - \frac{m}{2a},$$ for if $$r$$ be very great and $$v$$ small, the semi-major axis a must also be very great, and the path approaches the curve for which $$2a$$ is infinite, or a parabola.

This is the path described by a particle moving near the surface of the earth under the earth's attraction. The force which acts on the particle is really variable and directed toward the earth's centre, but within the limits of the path it may be considered constant and directed vertically downward.

This motion was first discussed by Galileo in connection with his study of falling bodies. His method was as follows: Let us assume the rectangular coordinates $$x$$ and $$y$$, of which $$x$$ is horizontal and $$y$$ vertical, drawn upward in the direction opposite to the acting force. Let a particle be projected from the origin in the plane of the axes with the velocity $$v$$ in a direction which makes with the $$x$$-axis the angle $$\alpha$$.

The component velocities along the two axes are then $$v \cos{\alpha}$$ and $$v \sin{\alpha}$$. At the end of any time $$t$$ reckoned from the instant at which the particle leaves the origin, the displacement of the