170 WORK 



140. THEOREM. During the motion of a particle under any sys- 

 tem of forces, the increase in kinetic energy is equal to the total 

 work done on the particle by external agencies. 



Let us consider motion of a particle from one position P to a 

 second position Q, and let the velocities of the particle at these 

 two points be v p , V Q respectively. 



Let us examine the motion over any element ds of the path, 

 and let the velocities at the beginning and end of this element be 

 v and v + dv. Let P be the force, or component of force along ds, 

 which acts on the particle while it describes the element ds of its 

 path. If dt is the time taken to describe this element of path, the 



ci/ij 



acceleration is > and since the force acting in the direction of 

 at 



motion is P, we have, by the second law of motion, 



dv 



P = m 

 dt 



SjftJ 



Hence, as in 139, Pds = m ~ ds 



at 



da' 



= m dv 

 dt 



= mv dv. 



Integrating over the whole path from P to Q, we obtain 



/Q /<? 



I Pds = m I vdv 

 Jp JP 



= $mv*-$mv* (37) 



= increase in kinetic energy. 



The left-hand side of this equation represents the work done on 

 the particle, proving the result required. 



141. The work performed on the particle by external forces may 

 be regarded also as equal to minus the work performed by the parti- 

 cle on external agencies. For if P is the force acting on the particle 

 along ds, it follows from the equality of action and reaction that 

 the force acting on the external agencies from the particle is P, 



