MECHANICS. 



eCXJo 



Fig. 102. 



directions, leuvo them free to fall together and Mtriko each other. 

 Since both fall from the same height, their velocities are equal, 

 and they each have the aamo mosg ; their momenta are there* 

 fore equal, and being in opposite directions neutralise each other. 

 Both bulls will therefore, after impact, remain at rest. 



In order to measure the distance through which the balls fall, 

 we must draw the arcs which they describe and divide them. 

 We do not, however, make the divisions equal, but draw a series 

 of parallel lines at tho same distance apart, the lowest being 

 even with tho top of tho balls, and make our divisions at tho 

 points where those cut the arcs. Tho reason of this is, that tho 

 Telocity is proportional, not to tho length of arc, but to the 

 vertical height, and thus those divisions indicate the velocity. 



Now raise c to the fourth division, and let it fall against n. 

 No momentum will be destroyed; it will merely bo shared 

 between the two balls, as much being gained by tho one as is lost 

 by the other ; and, since both bolls have the same weight, each 

 will move with half the velocity that c had OB striking D. They 

 will therefore rise to- 

 gether to the first divi- 

 sion of tho arc D F, for 

 C takes twice as long 

 to fall from 4 as it 

 does from 1 , and the ve- 

 locity is proportional 

 to the time, therefore 

 it acquires a double 

 velocity in falling. 

 Now whatever velo- 

 city a body acquires 

 in falling from any 

 height, it must start 

 with that velocity to rise to that height. A velocity, then, half 

 as great as that acquired by c will raise the two balls to 1. 



In the same way, if we make c half as heavy as D, and raising 

 it to the 9th division let it fall, the two will, as before, rise to 1. 

 The mass moved after impact is three times that of c, the 

 velocity will therefore be only one-third as great; they will 

 therefore rise J the height. Wo see thus that when one body 

 strikes against another, the momentum will be divided between 

 them, and hence the resulting velocity will bo as much less than 

 that of the moving body as the mass of the two is greater than 

 its mass. 



For example, suppose a ball weighing 1 Ib. and moving with a 

 velocity of 60, to impinge against a larger ball weighing 141bs. 

 The mass after impact will be 15 Ibs., or fifteen times that of the 

 ball ; tho velocity will therefore be ^ 3 , or 4 feet per second. No 

 momentum is lost. The original momentum was 1 x 60 ; .after 

 impact it is 15x4, which is also equal to 60. 



This principle supplies us with a means of measuring very 

 great velocities, as that of a cannon-ball or other missile. 



A large block of wood or metal is suspended by a rod so as to 

 swing to and fro with as little friction as possible. This is 

 called a ballistic pendulum. Against this the boll is caused to 

 strike, and by its impact it sets it in motion. A graduated arc 

 is fixed under the block on which tho distance to which it 

 swings can be noted, and from this wo can calculate the velocity 

 it had immediately after the ball struck it. We have only to 

 measure the vertical height to which it rose, and ascertain the 

 velocity it would attain in falling from that height, and thus 

 we have -the velocity with which it started. 



The weight of the bullet and the pendulum being also known, 

 we can at once determine tho proportion they bear to each other, 

 and thus wo can ascertain the velocity of the ball from that of 

 the pendulum. 



Suppose, for example, that the pendulum weighs half a ton. 

 and being struck by a ball weighing 24 Ibs. is raised to a height 

 of 16 feet. In falling from this height it would acquire a 

 velocity of 32 ; this, therefore, is that which it had immediately 

 after the boll struck it. But tho mass of the ball is to that of 

 tho two together as 24 to 1144, or 1 to 48 nearly. Tho velocity 

 of the ball was therefore 48 x 32, or 1536 feet per second. 



Hence wo see why, if one body strikes against another, the 

 heavier it is as compared with that against which it strikes, tho 

 greater tho effect produced. If wo want to drive a large nail or 

 to strike a violent blow, we use a heavy hammer, for by it we 

 obtain a much greater momentum, and thus accomplish the 

 work with greater ease. So, too, when we are driving a nail 



into a plank, we place a support bohind or hold a beary hammer 

 against it. Unions we do thu the momentum is shared by the 

 board, which yield* to the blow, and thn* destroys much of the 

 effect But when a heavy inelastic body is held behind, thu, 

 too, hai to share the momentum, and thus the plank yield* 

 much less, and the nail is driven more easily. 



In tho same way tome of the feaU of strength sometimes 

 exhibited may bo explained. A man will lie with his shoaldsn 

 supported on one chair and his feet on a second, A heavy anvil 

 is then placed on hi body, and on thu he allows stones to be 

 broken or blows to bo struck, which, bat for the anvil, most 

 certainly kill him. Tho reason is, that the momentum of the 

 hammer imparts but a very slight relooity to the anvil, on 

 account of the greatly superior weight of the latter. This 

 small velocity is easily overcome by the muscled, which being 

 stretched, act, to a certain extent, like a spring, and thus the 

 blow Is scarcely felt. 



We must now pass on to consider the impact of elastic 

 bodies, and for this we may take balls of ivory suspended in the 

 same way as those of clay were. We shall find that, though the 

 effects produced by these are different, the same general laws 

 apply. The bodies, howov r, instead of moving on together, 

 will, after impact, rebound and fly apart 



Let us raise one of the balls c (Fig. 102) and allow it to fall 

 against tho other. The first effect will be that the momentum 

 will be shared between the two, but, being elastic, they will be 

 compressed, and tho reaction in regaining their shape, being 

 equal and opposite to tho action, will destroy the motion of c and 

 double that of D. The former will therefore remain at rest, and 

 D will move on with a velocity equal to that which c had. If a 

 series of several balls be thus suspended so as just to touch one 

 another, and the end one raised and allowed to fall against the 

 others, the motion of the first will be imparted to the second, 

 and by that to the third, and so on throughout the entire series, 

 the motion of each being destroyed by the reaction of the next 

 The result will thus bo that tho end ball only will rise, all the 

 others remaining at rest. So, if two balls be allowed to fall, 

 two will be raised at the other end. We see, then, that no 

 momentum is lost hero, any mure than it was in the case of in- 

 elastic bodies ; but it is not shared between all the balls, as it 

 was in the other case. These experiments con, of course, be varied 

 to almost any extent, and you are recommended to try them for 

 yourselves, for more is always learnt by seeing or trying a few ex- 

 periments than by reading about many. As, however, there is 

 difficulty in procuring and suspending ivory balls, the experiments 

 can be tried in a simpler way with common glass marbles. Lay 

 two thin strips of wood along a smooth surface, like the top of s 

 table, and adjust their distance so that a marble may just roll 

 along between them; or, better still, cut a small groove in which 

 the marbles may run. One marble may then be laid in the 

 groove, and another made to strike it gently. The latter will 

 come almost to rest, while the other will move. The reason 

 why it does not come absolutely to rest is, that glass is not 

 perfectly elastic, and thus the reaction is not 

 quite sufficient to destroy tho motion. If seve- 

 ral marbles be laid so as to touch one another, 

 and one made to strike the end, the same re* 

 suits will ensue as with the ivory balls. 



There is one other law relating to impact 

 It is, that "the angle of incidence is equal to the 

 Fig. 103. angle of reflection." The meaning of this will 

 be clear from the annexed figure. Let any 

 body strike against a surface A c, in the dire<tion D B, it will re- 

 bound from it in the direction B E, making the same angle with 

 the perpendicular B V that B D does. The angle D B r, or that 

 at which it strikes A c, is called the angle of incidence, while 

 r B E is the angle of reflection, and the law asserts that these 

 are always equal. As we pass to optics and other branches of 

 physics, we shall find further illustrations of this law. 



ANSWERS TO EXAMPLES IN LKSSON 



1. It will rise a little orer 156 feet, and will reach tho earth again in 

 *i seconds. 



2. The elevation is 5J 1 x 16, which aqua* SOI x 16, or 484 faet 



3. It will strike the earth with a velocity of 160. 



4. It will take 7 seconds, in the last o! which it will fall 308 feet 



5. 16 x 33, or 523 feet 



6. It would require S seconds, and pass over 576 feet 



