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COLLISION, IMPACT. 



COLLISION, IMPACT. 



23 



equally evident, that if, after the above correction is made, the object 

 end o appears too high by n", that the true angle with the vertical is 

 90 + n", or that the reading of the circle should show n" of depres- 

 sion. The different cases which may occur present no difficulty. If 

 the collars are truly cylindrical and the level a delicate one, such a 

 collimator should show the true horizontal point within 1". The 

 telescope should not be very small, not less than 12 inches. 



It would scarcely be just not to notice under this head an instru- 

 ment by Roe'mer, which has as much merit, as an invention, as any of 

 these which we have described. It consists of two equal lenses fixed 

 in a tube at a distance somewhat exceeding their focal length, with a 

 system of wires in the focus of each, between the glasses. By applying 

 the proper eye-piece at each end, the near wires, and consequently 

 objects through the most distant object-glass, are made visible. The 

 two object-glasses and the crosses of the wires being all adjusted in 

 the same straight line, it is evident that, on looking in at each end of 

 the tube, objects 180 apart will be seen on the crosses. Roe'mer called 

 this tube an amphioptron, or reciprocal tdecope, and used it for the 

 transit adjustment in collimation of his rota meridiana. (Horrebow, 

 ' Basis Astronomise,' p. 97.) 



For further details, see Pearson's ' Practical Astronomy,' vol. ii. p. 

 446, plate xxi. 



COLLISION, IMPACT, or PERCUSSION OF BODIES, is that 

 part of Dynamics in which are contemplated the effects arising from 

 the striking against each other of two bodies, one or both of which are 

 in motion, and answers to the choc de corps of French treatises. 



It is usual to treat the first principles of this subject by supposing 

 the bodies in question to be spherical ; and for the following reason : 

 When a body receives a blow, if it be free to turn as well as to move 

 forward, a rotatory motion is, generally speaking, produced, as well as 

 a motion of translation. But if the direction of the blow be in a line 

 which passes through the centre of gravity, no rotatory motion is pro- 

 duced. Now if two equal spheres move upon a plane, it is obvious 

 that when either strikes the other, the direction} of the blow passes 

 through the centre of gravity. Making use then of equal spherical 

 balls, of the same or different weights, moving upon a level plane, let 

 it be remembered that all conclusions apply equally to bodies of any 

 form, having no rotatory motion, and striking each other in such a way 

 that the line joining their centres of gravity passes through the point 

 of contact at the moment when they strike. 



The simple mathematical theory of impact proceeds, like other 

 mechanical theories, upon suppositions which can only be approxi- 

 mately obtained in practice. For instance, if in the preceding supposi- 

 tion the level plane and the balls exercise any friction on each other, 

 the consequence will be that the balls will begin to roll on the table, 

 even though the blows which set them in motion pass through their 

 centres. To the existence of this friction are due many phenomena 

 which a game of billiards will present, and which will not result from 

 the common theory. Let the table, then, be supposed to exert no 

 friction on the balls, so that one of the latter, struck by a blow the 

 direction of which passes through the centre, wUl more along the table 

 without rolling. 



Let us now suppose the ball A to be impelled directly towards an 

 immovable obstacle, such as an upright ledge at the end of the table. 

 On striking this ledge, the ball will, generally speaking, recoil more or 

 legs. Some substances will hardly give any recoil, while others will 

 send the ball back with nearly the same velocity as that of its approach. 

 This spring or elasticity is more easily measured than explained ; it 

 arises in the following manner : At the moment of impact, the ball 

 compresses the part of the obstacle against which it strikes, which 

 pressure continues until the reaction of the obstacle has destroyed all 

 locity of the ball. At the same time the parts of the ball close 

 to the point of impact have been compressed in a similar manner. If 

 then there were no effort in the parts of the obstacle nor in those of 

 the ball to recover their former position, the ball would remain at rest, 

 close to the obstacle. If the recoil were complete, that is, if the parts 

 of both bodies endeavoured to recover their position with a force equal 

 to that which disturbed them, the recoil would rapidly but gradually 

 create in the ball a velocity equal to that with which it approached. 

 These two cases are the theoretical extremes which it is most probable 

 no material bodies attain : in the first case they are said to be wholly 

 inelastic, and in the second the elasticity is said to be perfect. But if 

 only a fraction of the velocity of approach be restored, then e is said 

 to be the measure of the elasticity of the bodies. 



In treating of the theoretical effects of impact, many authors have 

 ascribed to bodies the hypothetical property of perfect hardness or 

 incompressibility. This, however, is quite gratuitous, for Mr. Hodg- 

 i has not found in the course of his experiments (see ' British 

 Association Reports,' vol. iii. p. 534), any matter perfectly fulfilling 

 these conditions. Hence the value of e for all known substances is a 

 positive proper fraction, which represents the ratio that the force of 

 restitution licars to the compressing force, that is, 



force of restitution. 



, force of compression. 



This quantity must not be confounded with the madului of elai- 

 [KLASTICITT.] 



The value of e for some common substances is as follows : 



Glass = 0-94 



Ivory = 0-81 



Steel = 0-79 



Cast-iron = 0'73 



Bell-metal = 0-67 



Cork = 0-65 



Brass = 0-41 



Lead = 0-80 



If the striking bodies have spherical forms so that the contact may 

 take place, at the first instant, in a point, their surfaces about that 

 point will have their figures changed ; and if the bodies have different 

 degrees of hardness, an indentation may take place in that which is 

 the least hard, the other penetrating to a certain distance in it. When 

 the bodies are soft, like balls of wet clay, the change of figure produced 

 by collision is manifest ; but when two balls possess an elasticity which 

 is nearly perfect, they so far recover their original figure after impact, 

 that the change is not perceptible : it may be rendered evident, how- 

 ever, by covering one of the spheres with ink and suffering it to 

 impinge on the other, the latter then receives a stain which, instead of 

 being a point, is a circle of sensible magnitude ; and this proves that 

 the surfaces must have been flattened at the point of impact. 



Impact has been termed a " pressure of short duration ; " but practi- 

 cally there is a great difference between the effects of pressure and 

 impact : thus, a very large weight will be required to press a nail into 

 a block of wood, which may be readily driven into it by a small ham- 

 mer ; and the reason is, that a longitudinal compression of the nail 

 towards the head takes place, which is followed by restitution, and 

 these actions follow each other successively, and the nail enters by a 

 kind of vermicular action, like that of a worm progressing through the 

 earth. 



Also, when impact is employed to communicate motion to one body 

 relatively to another, the effect produced depends greatly on the immo- 

 bility of the latter ; thus, many more blows will be required to drive 

 a nail into a loose board, than would suffice if the latter were fixed ; 

 although, in certain cases, as in the breaking of minerals, &c., the effect 

 of impact is diminished by a firm support. Here, probably, the effect 

 of momentum on the successive particles is interfered with by a con- 

 trary momentum generated by restitution. 



lu order to account for the effect of percussion in impelling a body, 

 a wedge for example, being much greater than that of mere pressure, 

 it may be observed that both effects depend ou the product of the mass 

 of the impelling body and its velocity : but, when a body moves in 

 consequence of pressure, the velocity is extremely small ; therefore, hi 

 order that the effect of simple pressure may be equal to that of per- 

 cussion, the mass imposed must be very great. It is evident, however, 

 from what has been said, that the two forces are of the same nature. 

 It should be added, here, that the shock produced in a material, when 

 divided by a wedge, or penetrated by a nail, either of these being driven 

 with a force produced by a sudden blow of a hammer may, by dis- 

 placing the particles of the material, diminish their cohesive power ; 

 and this may be, in part, the reason that the effect of percussion often 

 exceeds that of a weight many hundred times greater than that of the 

 hammer. 



The force of elasticity is very different iff different bodies : spheres 

 of glass are those in which the force of restitution (after impact) 

 approaches nearest to the force of compression ; and, in such spheres, 

 the ratio between the forces is as 15 to 16: in spheres of ivory the 

 ratio is as 8 to 9 ; and in spheres of steel, as 5 to 9. 



The bodies upon which experiments on collision are usually made 

 are generally of a spherical form ; in order that when they impinge 

 upon one another directly it may be indifferent at what part of the 

 surfaces of the bodies the contact takes place : the bodies are usually 

 suspended by a string or rod from fixed points ; and they are made to 

 impinge upon one another while describing circular arcs, in a vertical 

 plane, about the point of suspension. The absolute momentum, or 

 quantity of motion in a body, is represented by the product of its mass 

 and the velocity with which it is moving : but the effects of the col- 

 lision of two bodies depend on their relative velocity, or that with 

 which they approach to, or move from, one another; this is con- 

 sequently the sum of the absolute velocities when the bodies, in 

 approaching each other, move in opposite directions, and the difference 

 when they move in the same direction. 



The principles upon which are determined the velocities .after impact 

 of different balls which strike one another are as follow : 



1. If two perfectly inelastic balls move towards each other in 

 opposite directions, and with velocities inversely proportional to their 

 weights or masses, they will destroy each other's velocities and remain 

 at rest. Thus if A were twice as heavy as B, but if B moved twice as 

 fast as A, there would be no motion after impact. [MOMENTUM; 

 MOTION, LAWS OF.] Let a be the velocity of A, and b of B ; then A 

 and B being expressed in the same unite of weight, and a and b in the 

 same unite of length and time, the preceding condition is fulfilled when 

 Aa = B&. 



2. If the same velocities be added to or taken from both balls, so 

 that their rate of approach is not altered, the forces exerted in the 

 shock will not be altered, nor will the rate of recess after the shock. 

 Thus a cannon-ball rebounding from a wall, both having the motion of 

 the earth, strikes with the same force and rebounds in the same manner 

 as it would do if the motion of the earth were taken from both, or if 

 the earth were at rest. 



