270 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



The impulses exerted on the body can be expressed as a single 

 impulse at any origin and an impulsive couple. 



The equations of impulsive motion express the equivalence of 

 the two systems of vectors. 



Thus if the impulses are reduced to an impulse at the centre of 

 inertia, whose resolved parts in the specified directions are X and 

 F, and a couple N, we can take the equations of motion to be 



m ( u - U) = X, m (v-V) = Y, mk* (o&amp;gt; - fl) = N. 



More generally, the resolved part, in any direction of the vector 

 whose resolved parts, in the specified directions, are m (u U) 

 and m (v V) is equal to the resolved part, in the same direction, 

 of the vector whose resolved parts, in the specified directions, are 

 X and F; and the moment about any axis of the vector system 

 determined by m (u U), m (v V), mk* (o&amp;gt; fl), is equal to the 

 moment about the same axis of the vector system determined by 

 Z, F, N. 



*238. Kinetic energy produced by impulses applied to 

 rigid body. Suppose the body to move in one plane. Let m be 

 the mass of the body, U, V resolved velocities of its centre of 

 inertia parallel to the axes of reference, and H its angular velocity 

 just before the impulses act, u, v, &&amp;gt; corresponding quantities just 

 after. 



Let X lf Fj be the resolved parts parallel to the axes of the 

 impulse applied to the body at any point whose coordinates relative 



to the centre of inertia are as lt y a . 



Y \ 

 The equations of impulsive motion are 



mk* (w - H) = 2 (xY-yX). 

 Multiply these equations in order by 



and let T be the kinetic energy of the body after the impulses, T 

 that before. Then we have 



T - T = 2 1 { X (u - toy + U - %) + F (v + tox + F + 



The right-hand member of this equation is the sum of the 



