16 KINETICS OF A PARTICLE. [32. 



The components of the velocities along CO must satisfy the 

 equations (7) and (10). Hence, substituting u cos a, ' cos a', 

 v cos ft z/ cos/3' for z/, V, z/, z/, respectively, we must have 



mv cos ft + m'v f cos @' = mu cos a + m'ti' cos ', (19) 

 z/ cos /3' v cos ^ e(u cos a u f cos a'). (20) 



32. The particular case of the oblique impact of a homoge- 

 neous sphere against a smooth fixed plane deserves special 

 mention. In this case, ' and v' are zero ; and the angles a, /3 

 made by the velocities u, v with the normal to the plane, are 

 called the angle of incidence and of reflection, respectively. 



The equations (18) and (20) reduce to the following: 



v sin j3 = u sin a, v cos /3= eu cos a, (21) 



where the minus sign indicates that the projections of u and v 

 on the normal have opposite sense. Dividing the former of 

 these equations by the latter, we find 



tan a= e tan ft (22) 



where the minus sign merely indicates that the angles a, /3 lie 

 on opposite sides of the normal. 



For perfectly elastic bodies, the last equation shows that the 

 angles of incidence and reflection are equal. 



33. Exercises. 



(1) A baseball weighing 5^ oz., while moving with a velocity of 100 

 ft. per second is struck by the bat in a direction at right angles to its 

 line of motion. Find the momentum imparted by the blow if it deflects 

 the ball through an angle of 60. 



(2) Determine the velocity of recoil of a gun weighing 1500 Ibs. when 

 a i2-lb. shot is fired from it with an initial velocity of 2000 ft. per 

 second. 



(3) The heavier one of two ivory balls (e = 0.88), whose centroids 

 are C, O and whose masses are i Ib. and j lb., impinges upon the 

 lighter. The velocity of the heavier ball is 15 ft. per second and makes 

 an angle of 30 with the line CO, while the velocity of the lighter ball is 

 5 ft. per second and makes an angle of 60 with the line CO (pro- 

 duced). Find the velocities after impact in magnitude and direction. 



