226 MOTION OF SYSTEMS OF PARTICLES 



These laws may be regarded as the extensions of Newton's Laws 

 I and II to the motion of a system of particles. We can see now 

 why it is often legitimate to apply Newton's second law to the 

 motion of bodies of finite size, as though they were particles 

 (cf. 26). 



The principle of conservation of momentum is often sufficient 

 in itself to supply the solution of a dynamical problem in which 

 only two bodies are in motion. 



ILLUSTRATIVE EXAMPLE 



A shot of mass m is fired from a gun of mass M, which is free to run back on 

 a pair of horizontal rails. Find the velocity of recoil of the gun, and examine the 

 influence of the recoil on the motion of the shot. 



Let us suppose that, before firing, the gun stands pointing at an angle a to the 

 horizon, and let the muzzle velocity of the shot i.e. the velocity relative to 

 the gun with which the shot emerges be V. 



Let us suppose that the velocity of the shot relative to the earth has compo- 

 nents w, v horizontal and vertical, and let the velocity of recoil of the gun be U, 

 measured in the horizontal direction opposite to that in which the gun is pointing. 



The system consisting of the gun, powder, and shot is not free from the 

 action of external forces, but these forces, namely the weight of the system and 

 its reaction with the earth, have no horizontal component. Thus the horizontal 

 momentum of the system must remain unaltered by the explosion. This hori- 

 zontal momentum was zero initially : it is therefore zero when the shot leaves 

 the gun. Thus we have, neglecting the weight of the powder, 



M U -mu = Q. (a) 



The velocity of the shot relative to the gun has components 



u + U, v. 



This velocity must, however, be a velocity V making an angle a with the 

 horizontal. We therefore have 



u + U = V cos a, (6) 



v = V sin a. (c) 



From equations (a) and (6) we find 



u _ U _ V cos a 

 M~ m ~ M -f m 



Thus the velocity of recoil is 



m 

 T^coscr. 



