6 KINETICS OF A PARTICLE. [12. 



u, u* to have the same sense and u>u', so that m will finally 

 impinge upon m'. The case when the velocities are of oppo- 

 site sense will not require special investigation, as only the 

 sign of u' would have to be changed. 



// is our object to determine the velocities v, v' of m, m', im- 

 mediately after impact, when the velocities _, ' immediately 

 before impact are given. 



The results here derived for homogeneous spheres hold, 

 generally, whatever the shape of the impinging bodies, provided 

 that they do not rotate, and that the common normal at the 

 point of contact passes through both centroids. 



12. If the spheres were perfectly rigid, the problem would be 

 indeterminate, for there is no way of deciding how the velocities 

 would be affected by the collision. 



Natural bodies are not perfectly rigid. The effect of the 

 impact will, in general, consist in a compression of the portions 

 of the bodies brought into contact. Moreover, all natural bodies 

 possess a certain degree of elasticity ; the compression will 

 therefore be followed by an extension, each sphere tending to 

 regain its shape at least partially. 



The compression acts as a retarding force on the impinging 

 sphere m, and as an accelerating force on m' . It will last 

 until, the velocities #, u 1 have become equal, say =w. During 

 the subsequent period of extension, or restitution, the elastic 

 stress still further diminishes the velocity of m y and increases 

 that of m', until they become, say, v, v'. 



13. The stress varies, of course, during the whole time r of 

 compression and restitution. But, according to Newton's third 

 law, the pressure F exerted at any instant by m on m' must be 

 equal and opposite to the pressure F' exerted by m' on m at 

 the same instant. Since F= mdu/dt, F 1 = m'du'/dt, and F= - F 1 

 at any instant during the time r, we have 



, or m f r du = -m' Cdu', 



t/o JQ 



