Page:Elementary Text-book of Physics (Anthony, 1897).djvu/52

38 certain limitations assumed for the system, we are able fully to describe its motions. The first of these is that of a pair of bodies which act on each other with a force, the direction of which is in the line joining the bodies. This case, known as the problem of two bodies, may be completely solved. The problem of three bodies can be solved only approximately, under certain limitations as to the relative magnitudes of the bodies. The second case is that in which the system forms a rigid body. While no truly rigid bodies exist in Nature, yet the changes of shape which most solids undergo under the action of ordinary forces are so slight in comparison with their dimensions that in many cases we may consider such solids as rigid, and illustrate the theorems relating to rigid bodies by experiments made upon solids. We shall first examine the motion of rigid bodies, and we shall limit ourselves to the case in which the motions of any one particle of the body always take place in one plane. By thus restricting the problem, it is possible to obtain the most essential facts connected with the motions of rigid bodies without the use of advanced mathematical methods.

35. Impact.— The changes in motion impressed upon bodies by their impact with others depend upon so many conditions that they present complications which render the discussion of them impossible in this book. We will consider, however, the simple case of the impact of two spheres, the centres of which are moving in the same straight line. We call the masses of the two spheres $$m_{1}$$ and $$m_{2}$$ and their respective velocities $$u_{1}$$ and $$u_{2}$$. The two spheres constitute a system for which the velocity of the centre of mass is given by The bodies on impact are momentarily distorted, and a force arises between them tending to separate them, the magnitude of which depends upon the elasticity of the bodies. The velocity of the centre of mass will remain uniform, whatever be the forces acting between the bodies, and the momenta of the two bodies relative to the centre of mass, both before and after impact, will be equal and opposite. Call the velocities of the bodies after impact $$v_{1}$$ and