CONSERVATION OF ENERGY 173 



146. The theorem of 145 is clearly true when the particle is 

 ascending, in which case li is negative or if the particle ascends 

 during part of its path and descends during the remainder. More- 

 over, the particle may move under any conservative system of 

 forces, provided only that the whole potential energy arises from 

 the weight of the particle, and the theorem remains true. 



It is true, for instance, of a particle tied to an inextensible 

 string, or of a particle moving 'freely in a vacuum. 



To illustrate the use of the theorem, let us suppose that a bicyclist, 

 riding with a velocity of 15 miles an hour, comes to the top of a hill of 

 height 60 feet, down which he coasts. Let us find his velocity at the 

 bottom, on the supposition that friction, air resistance, etc., may be 

 neglected. 



Taking the top and bottom of the hill to be the points P, Q respectively 

 of the theorem just proved, we have, from the data of the problem, 



h = 60 feet, 



u = 15 miles per hour = 22 feet per second. 



Thus, using foot-second units, we have 



y 2 = w 2 + 2 gh = 22 2 + 2-32-60 = 4324, 

 so that v = 66 feet per second, approximately, 



= 45 miles per hour. 



Thus the velocity of the bicycle, if unchecked by friction or air resist- 

 ance, would be one of about 45 miles per hour. 



EXAMPLES 



1. An automobile running 40 miles an hour comes to the foot of a steep hill, 

 and at the same instant the engine is shut off. To what height up the hill will 

 the automobile go before coming to rest (neglect friction, etc.)? 



2. A laborer has to send bricks to a bricklayer at a height of 10 feet. He 

 throws them up so that they reach the bricklayer with a velocity of 10 feet per 

 second. What proportion of his work could he save if he threw them so as only 

 just to reach the bricklayer ? 



3. A gun carriage of mass 3 tons recoils on a horizontal plane with a velocity 

 of 10 feet per second. Find the steady pressure that must be applied to it to 

 reduce it to rest in a distance of 3 feet. 



4. A ship of 2000 tons moving at 30 feet a minute is brought to rest by a 

 hawser in a distance of 2 feet. Find in tons what pull the hawser has to sustain. 



