3i.] IMPACT OF SPHERES. 15 



gun. Assuming the line of motion of the centroid of the shot 

 to pass through the centroid of the gun, we may apply equation 

 (7), with u = o, u' = o. Hence, denoting by m the mass of the 

 gun, by m' that of the shot, we find for the velocity of recoil 





(17) 



The kinetic energies ^ mv* and |- m'v'^ of gun and shot are in 

 the ratio m 1 /m ; hence, the energy of recoil is the fraction 

 m'/(m + m f ) of the total energy J mv* + ^ m'v' 2 of 'the explosion 

 of the powder, while the energy of the shot is =m/(m-}-m ! ) of 

 the total energy. In large guns the recoil is diminished by a 

 special elastic cushion or "compressor." Moreover, the mass 

 of the powder gases cannot be entirely neglected in all cases. 



31. Oblique Impact. In ihe case of oblique impact, i.e. when the 

 centres of the colliding spheres do not move in the same straight 

 line, the velocities after impact can be found without difficulty, 

 provided that the velocities of the centres before impact lie in 

 the same plane and that the spheres are perfectly smooth. 



With these assumptions, let m, m' be the masses of the two 

 spheres ; C, C' their centres (Fig. 3) ; u t u' the velocities before 

 impact ; a, a f the angles 

 made by ?/, u' with the line 

 CC' ; v, v' the velocities 

 after impact ; and 0, 0' 

 the angles they make 

 with CC. 



As there is no friction, 



the forces of impact act Fig. 3? 



along the line CC that 



joins the centres. Hence, resolving each velocity along and 

 perpendicular to CO, the components at right angles to CO 

 must remain unchanged by the collision ; that is, we must have 



v sin/3=& sin a, v' sin' = ' sin a', (18) 



