KINETIC ENERGY 229 



Since, by the third law of motion, no force can act singly, it fol- 

 lows that, in every case of calculation of work from kinetic energy, 

 it will be necessary to consider the kinetic energy of more than one 

 body. For instance, a man throwing a ball on land will not only 

 jerk the ball forward, but will also jerk the whole earth backward, 

 and the energy of both must be taken into account, or we shall get 

 erroneous results. 



185. A second difficulty, closely connected with the first, suggests 

 itself at once. Suppose we have a ball thrown with a velocity v 

 along the deck of a ship moving with velocity V. We have seen 

 that we must not suppose the kinetic energy of the ball to be 

 \ mv*, but is it any more legitimate to suppose it to be \ m (v + F) 2 ? 

 For the sea in which the ship sails will have a further velocity V' 

 in consequence of the earth's rotation, so that the energy might 

 equally well be taken to be 



and so we might go on indefinitely. Knowing of no frame of 

 reference which is absolutely at rest, it would seem to be impos- 

 sible to find the true value of the kinetic energy. Moreover, 

 it ought to be noticed that the expressions for the kinetic energy 

 referred to different frames of reference differ by more than mere 

 constants. For instance, the difference between the two expres- 

 sions we have found for kinetic energy relative to the sea and 

 kinetic energy relative to the earth's center is 



Jm(v+F+F / ) 2 -lm( 



This difference not only depends on m and V' but also on v and V. 

 It is not a constant difference, and so does not disappear when we 

 calculate the increase in kinetic energy resulting from the action 

 of forces. 



The theorems which follow serve the purpose of showing a way 

 through these and similar difficulties. 



