MECHANICS 721 joining the centres, and a speed t sina in a known direction perpendicular to this, namely, in the plane through this and its original direction of motion. And similarly for the sphere M . Thus the consequences of the impact are completely determined. 188. When a sphere of mass M impinges directly, with speed V, on another M at rest, the speed acquired by M is M V(l + e) M + M But, if another sphere of mass //,, also at rest, be interposed between them, M will acquire a speed This is greatest when /u. is the geometric mean of M and M , and its value is then MV(l+c)* The ratio of this to the speed which M would have acquired without the interposition of the third sphere is 1 + 2 VMM/ M + M There is thus a gain by the interposition if, and only if, This condition is always satisfied when the coefficient of restitution is unity, except in the special case of equal masses. If an infinite number of spheres be interposed between M and M , so adjusted as to give the greatest possible speed to M , that greatest speed is V ^/M / M, provided we have e= 1. Continuous Succession of Indefinitely Small Impacts. nfinite 189. We maynow consider the case of a continuous series eries of of indefinitely small impacts, whose effect is comparable with n ? mt that of a finite force. One obvious method of considering mpacts suc h a problem is to estimate separately the changes in the velocity produced by the finite forces and by the impacts, in the same indefinitely small time 8t, and compound these for the actual effect on the motion in that period. Another way, of course, is to equate the rate of increase of momentum per unit of time to the force producing it. A mass, under no forces, moves through a uniform cloud of little particles which are at rest. Those it meets adhere to it. Find the motion. At time t let /u. be the mass, and let x denote its position in its line of motion. Then, as there is no loss of momentum, we have
- 0*)-o.
But if M be the original mass, /* the mass of the particles picked up in unit of length, obviously 0, x = V, vhen t = ; and from which x can be easily found. It is interesting to observe that we have Substitute and integrate, supposing x we get Rocket. so that the mass moves as if acted on by an attraction varying in versely as the cube of the distance from a point in its line of motion. This problem obviously leads to the same result as the following : A cannon-ball attached to one end of a chain, which is coiled up on a smooth horizontal plane, is projected alonr/ the plane. Determine its motion. 190. Another excellent instance of the application of this process is furnished by the motion of a rocket, where the motive power depends on the fact that a portion of the mass is detached with considerable relative velocity. The increase of the momentum of the rocket due to this cause is equal to the relative momentum with which the products of combustion escape. If we suppose the rocket, originally of mass M, to lose eM in unit of time, projected from it with relative velocity V, the gain of momentum in time 8t due to this cause is The total upward acceleration is therefore cMV n . M cAlY Unless this be positive the rocket cannot rise. It will rise at once if V>g/e, and it cannot rise at all unless MV/M >#/e, M being the mass of the case, stick, &c., which are not burned away. From the above data it is easy to calculate that the greatest speed acquired during the flight (the resistance of the air being left out of account) is M ff/. ar Dynamics of a System of Particles Generally. 191. The laft of energy, in abstract dynamics, may be expressed as follows : the whole work done in any time, on any limited material system, by applied forces, is equal to the whole effect in the forms of potential and kinetic energy produced in the system, together with the work lost in friction. This principle may be regarded as compre hending the whole of abstract dynamics, because the con ditions of equilibrium and of motion, in every possible case, may be derived from it. 192. A material system, whose relative motions are unresisted by friction, is in equilibrium in any configuration if, and is not in equilibrium unless, the rate at which the applied forces perform work at the instant of passing through it is equal to that at which potential energy is gained, in every possible motion through that configuration. This is the celebrated principle of "virtual velocities," which Lagrange made the basis of his Mecanique Analy- tique. 193. To prove it, we have first to remark that the system cannot possibly move away from any particular configuration except by work being done upon it by the forces to which it is subject ; it is therefore in equilibrium if the stated condition is fulfilled. To ascertain that nothing less than this condition can secure the equilibrium, let us first consider a system having only one degree of freedom to move. Whatever forces act on the whole system, we may always hold it in equilibrium by a single force applied to any one point of the system in its line of motion, opposite to the direction in which it tends to move, and of such magnitude that, in any infinitely small motion in either direction, it shall resist or shall do as much work as the other forces, whether applied or internal, altogether do or resist. Now, by the principle of super position of forces in equilibrium, we might, without altering their effect, apply to any one point of the system such a force as we have just seen would hold the system in equili brium, and another force equal and opposite to it. All the other forces being balanced by one of these two, they and it might again, by the principle of superposition of forces in equilibrium, be removed ; and therefore the whole set of given forces would produce the same effect, whether for equilibrium or for motion, as the single force which is left acting alone. This single force, since it is in a line in which the point of its application is free to move, must move the system. Hence the given forces, to which the single force has been proved equivalent, cannot possibly be XV. 91 Motion of a sys tem of particles. Condi tion of equi librium. Virtual
velocities.