56 Mr. G. F. C. Searle on the Impulsive 



directions o£ u and w. For convenience, we write 



(31) 



The relations which w l5 u 2 , and a Fig. 3 . 



bear to u, w, and ylr are shown in 

 fig. 3. 



From the three equations 



to 2 = u 2 + u 2 — 2 u x u 2 cos a, 



t^ 2 = u 2 + £ie 2 — wzy cos yfr, 



U 2 2 = U 2 + \w 2 -f UW COS i/r, 



we obtain 



UiU 2 cos a = u 2 — J zy 2 , 



or n^i 2 cos u = n 2 — ^m 2 . 



Writing down two equivalent expressions for the area of 

 the parallelogram, we have 



u x u 2 sin a = uw sin y, 

 or 



n x n 2 sin a = nm sin i/r. 

 Hence 



v?w 2 —u^u 2 2 sin 2 a = w; 2 ('U 2 — w 2 sin 2 ^), 

 or 



m 2 — % 2 /z 2 2 sin 2 a =s m 2 (l — ft 2 sin 2 n|r). 



When these values are inserted in the results of § 11, we 

 obtain 



W l-n 2 + jm 2 l-n 2 + jm 2 + m(l-n 2 sm 2 ^ _ 



U "m(l-7i 2 sin 2 f )* l0g l~^ + i m 2 -m(l-/j 2 sin 2 ^)i ' 



or 



W_ v 2 ^-u 2 + \w 2 , v 2 — u 2 + \w 2 -\- w(y 2 — u 2 sm 2 yjr)^ ^ 



TJo~~w(v 2 — M 2 sin 2 i/r)2 ® # 2 — w 2 + ^tt; 2 — w(u 2 — w 2 sin 2 -\/r)£ 



§ 13. We now pass on to calculate the momentum radiated 

 in the pulse generated by an impulsive change of velocity of 

 the sphere. It will not be necessary to employ the integral 

 calculus, since the momentum can be deduced on dynamical 

 principles from quantities which are already known. 



To change the velocity of the sphere impulsively from u x 

 to u 2 it is necessary to apply a force F 12 to the sphere as long- 

 as the pulse is passing over the sphere. When the pulse is 

 clear of the sphere the force is no longer required. While 

 the pulse is passing over the sphere, P 12 probably changes 

 both in direction and in magnitude, but we are not concerned 



