After this the particle begins to return, the resistance therefore tends to increase x, and the equation of motion is Let U be the velocity with which the particle will return to the point of projection; then, putting x = 0 in the latter equation, we obtain = This shews, as we might expect, that the particle returns to the point of projection with diminished velocity. 212. The results of the last Proposition are applicable to bodies projected in a resisting medium vertically upwards or downwards under gravity; for the acceleration due to gravity may still be considered constant, although not the same as for a particle in vacuo. The effective attraction of gravity is in fact the difference of the weights of the body and the fluid displaced, so that if a be the ratio of the density of the fluid displaced to that of the body, effective gravity =W (1 − a) = Mg (1 − a), where W and M are the weight and mass of the body, and therefore the acceleration caused by gravity=g (1-a). By substituting this for fin the results of § 211, we may obtain formulæ for the motion of bodies in a vertical direction under gravity. Hailstones and raindrops afford a good illustration of the Terminal Velocity indicated by the result of § 211. 213. To find the equations of motion, in a resisting medium, of a particle under any forces. Let x, y, z be the co-ordinates of the particle relative to an assumed system of rectangular axes, at the time t, and let X, Y, Z be the component accelerations, parallel to the axes, due to the forces acting on the particle. Then denoting by R the retardation due to the resistance, which lies in the tangent to the path described, and in a direction opposed to the motion, we have R These are the general equations of motion. In any particular case R will be given as a function of the density of the medium and the velocity of the particle, and particular methods will be necessary for obtaining the path of the particle and its position at any time. These equations will enable us, when X, Y, Z are given, to determine the resistance that a given path may be described. 214. A particle under gravity is projected from a given point in a given direction with a given velocity, and moves in a uniform medium whose resistance varies as some power of the velocity; to determine the motion. Take the given point as origin, the axis of a horizontal, the axis of y vertically upwards, so that the plane of xy may contain the direction of projection; let g denote the acceleration of gravity, v the velocity of the particle at any point, u its horizontal component, & the inclination of the direction of motion to the horizon, and R = kv" the retardation due to the resistance. Then the equations of motion are, resolving horizontally and vertically, or, resolving in the direction of the tangent and normal, Integrating this equation, denoting by u, the velocity at the vertex of the trajectory, Equations (9), (10), (11) give t, x, y in terms of p. (10), .(11). and the integrals in (9), (10), (11) were calculated by quadratures for different values of y and for certain ranges of angle, and the nominal values tabulated, in Tables IV. V. VI. in Mr Bashforth's "Motion of Projectiles." |