232, 233] LOSS OF ENEKGY IN IMPACT. 263 



233. Deduction of Newton s Rule from a particular 

 assumption. Without assuming any law of restitution we may 

 throw the expression for the kinetic energy lost in impact into a 

 form depending only on the relative velocities along the line of 

 centres before and after impact. With the notation of the last 

 Article, we have 



JR_ _U-u _ u -U _ (U- U )-(u-u f ) 



mm m m m + m 



and the kinetic energy lost is 



__ , u _ _ , _ 

 2 m + m lv 



Hence, if W and W denote the relative velocities resolved parallel to 

 the line of centres before and after impact, the kinetic energy lost is 



_mm^ 

 2 m + m ^ 



or in words it is the product of one quarter of the harmonic mean 

 of the masses and the difference of the squares of these relative 

 velocities. 



The same result may be obtained from general principles. 



Since the motion of the centre of inertia of the spheres is 

 unaltered by the impact, and since the motions of the spheres at 

 right angles to the line of centres are also unaltered, we only 

 require the velocities of the spheres relative to the centre of inertia 

 resolved along the line of centres, the kinetic energy being that of 

 the whole mass moving with the centre of inertia together with 

 that of the relative motion (Article 104). 



Now the velocity of m relative to the centre of inertia, resolved 

 along the line of centres, is m W/(m + m) and the velocity of m 

 relative to the centre of inertia, resolved along the same line, is 



Just after the impact the velocities of m and m relative to the 

 centre of inertia, resolved in the same direction, are respectively 



m W Km + m?) and - mW l(m -f m ). 

 Hence the kinetic energy lost is 



/ I *2 &quot; V / 



^m+m ) \m -f m 



1 / m/ TIT- A 2 i &amp;gt; / ??l wA 2 

 - 4m , Tr Jm - , W , 



\m + m J \m+m / 



leading to the same result as before. 



