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WORK 



the elastic ; the stone moves under the accelerating influence of this tension, 

 and the tension of the elastic slackens. While this is in progress the stone 

 is acquiring kinetic energy, while the stretched elastic is losing potential 

 energy. By the theorem just proved, the kinetic energy gained by the 

 stone must be just equal to the potential energy lost by the elastic. 



When the Stone escapes from the^catapult, most of the potential energy 

 of the elastic will have disappeared, having been transformed into the 

 kinetic energy of the stone. After this a further transformation of energy 

 may take place while the stone is in motion. If the stone moves upwards, 

 its potential energy will increase, so that there must be a corresponding 

 decrease in its kinetic energy its speed must slacken. On the other 

 hand, if the stone moves downwards, the potential energy will decrease, 

 so that its kinetic energy will increase it will gain in velocity. 



145. A very important deduction from the principle of the con- 

 servation of energy is the following : 



THEOREM. If a particle slide along any smooth curve, being acted 

 on by no forces except gravity and the reaction with the curve, and 

 if u, v be the velocities at two points P, Q of its path, then 



v* = u*+2gh, (39) 



where h is the vertical distance of Q below P, i.e. is the vertical 

 projection of the path PQ described by the particle. 



Let h p , h Q denote the heights of P and Q above any horizontal 

 plane - for instance, the earth'a surface. 

 Then when the particle is at P its kinetic 

 energy is ^ mu*, and its potential energy 

 is mgh p . Thus its total energy is 



1- mu 2 + mgh p . 



Similarly at Q its total energy is 

 J- wy 2 + mgh Q . 



Since the system of forces acting is a 

 conservative system, the total energy remains unaltered. Thus 



1 mil? + mgh p = ^ mv* + mgh Q , 



so that v 2 if = 2 g (h p h Q ) = 2 gh, 



proving the theorem. 



