ivith reference to the Measure and Transfer of Force, 337 



§10. 



The condition of impinging bodies demonstrates the absurdity 



of assuming the motion of a body to be proportional to its 



velocity. 



Let us take a simple numerical example. A ball of 1 lb. 

 weight with velocity 9 strikes a spring interposed vipou another 

 ball of 3 lbs. weight which is at rest. The velocity of the centre 

 of gravity is 3 ; the motion of the 1 lb. ball relative to centre of 

 gravity is 6, and of the 2 lb. ball 3. These relative motions are 

 exactly reversed by the recoil of the spring after impact. Thus 

 the absolute motion of the 1 lb. ball after impact becomes 3, and 

 the absolute motion of the 2 lb. Imll 6. At the instant when, 

 the spring has obtained its maximum compression by the force 

 of the collision, the velocity of both balls is 3, the same as the 

 centre of gravity. 



Let us now compute the quantity of motion in the system, 

 assuming the motion of a body to be proportional to its velocity. 

 Before impact the amount is .... P''- x 9 + 2ibs. x = 9. 

 After the spring has received the force of 



compression, the amount is . . . . (!'''• + 2'^^-) x 3 = 9. 

 After recoil, when the spring has given 



out its force, the amount is ... . l^^- x 3 + 2ibs. x 6 = 15. 



Thus the spring has been bent, compressed, or as it wei'C, 

 wound up without taking any motion from the system, without 

 any cost of work ; and the system has actually gained motion by 

 the collision. The concurrence of the two bodies has nearly 

 doubled their collective momentum. 



If the motion of a body is assumed to be proportional to the 

 square of its velocity, the quantity of motion in the system before 



impactis pb- x 81 + 2ibs- x = 81. 



After the spring has received the force 



of compression, the amount is . . (l^^- + 2^bs-) x 9 = 27. 

 After recoil, when the spring has given 



out its force, the amount is . . . l^''- x 9 + 2"'s- x 36 = 8L 



The collective motion before and after impact is thus the same, 

 and 81 — 27 = 54 is the force that is taken to compress, bend, 

 or wind up the interposed spring : it is the work rendered latent 

 at the instant of impact but immediately restored. It is a dyna- 

 mic, not a static force, and with the same impinging bodies it is 

 proportional to the square of their relative velocity. Thus if the 

 2 lb. ball with velocity 1 meets the 1 lb. ball with velocity 8, 

 the relative velocity is still 9, and the force of impact still 54. 

 This is also apparent from the general expression for the force of 



impact, VIZ. V, —Va { — r - I . 



