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VI. 



IMPACT IN THREE DIMENSIONS. 

 By prof. M. W. J. FRY, M.A., F.T.C.D-. 



Read June 26, 1916. Published Febuuary 9, 1917. 



1. The problem of impact in three dimensions is incompletely discussed 

 by Roiith in his excellent treatise on Elementary Rigid Dynamics, of which 

 a seventh edition appeared in 1905, and which may be regarded as the 

 standard work on the subject. 



He does not show how the initial stage of the impact depends on the 

 roots of a certain equation F{6) = ; nor how generally the representative 

 point gets on to the line of no sliding ; nor that when sliding ceases and 

 rolling is impossible the motion of the representative point is along a line 

 determined by one particular root of F(0) = 0. Also, the solution he gives 

 of impact between perfectly rough bodies is not correct, as it may involve 

 the physical absurdity of supposing the impulsive normal reaction to be 

 negative. 



These points and others are discussed in this paper, and it is shown 

 that the course of the impact under the most general conditions in three 

 dimensions can be minutely traced, and lastly the correct solution of the 

 problem is given when the coefficient of friction is supposed to be very 

 great. 



2. The General Equations. — In order to deal with the problem of getting 

 the motion after impact, when a rigid body A strikes against a body A', 

 we trace the variation of the resultant blow delivered by A at the point of 

 contact 0, by resolving it into three components : — P and Q the components 

 along any two perpendicular axes drawn through in the common tangent 

 plane, and R the component along an axis drawn in the direction of the 

 common normal, so that it is initially and always positive. We follow the 

 movement of the extremity of the resultant blow, whose co-ordinates are 

 P, Q, R, and call it the representative point. 



For A, let M be its mass ; u, v, tv the components of the velocity of 

 its centre of gravity ; to^, wy, w~ the components of its angular velocity ; 

 U, V, W, Q.;,, Hy, S2s the initial values of the same quantities ; A, B, G its 



B.I.A. PROC, VOL. XXXIII., SECT. A. fl2] 



