the sign being taken according as the motions of the bodies before impact are both in the same direction or in opposite directions.

If the second body was at rest before impact, V,=0, and

The demonstration of this proposition is wholly indepen dent of any hypothesis as to the nature of the impinging bodies or their elastic properties; the proposition is therefore true of all bod'es, whatever may be their degrees of hardness or their elasticity, provided only that at the instant of greatest compression every part of each body partakes in the common velocities of the bodies, there being no relative or vibratory motion of the parts of either body among themselves.

437. TO DETERMINE THE WORK EXPENDED UPON PRODUCING

THE STATE OF THE GREATEST COMPRESSION OF THE
FACES OF THE BODIES.

SUR

The same notation being taken as before, the whole work accumulated in the bodies, before impact, is represented

W

W

by V+V,'; and the work accumuiated in them

g

g

at the period of greatest compression, when they move with

the common velocity V, is represented by

Now the difference between the amounts of work accumulated in the bodies in these two states of their motion has been expended in producing their compression; if, therefore, the amount of work thus expended be represented by 11, we have

or substituting for V its value from equation (701), and reducing,

This expression represents the amount of work permanently lost in the impact of two inelastic bodies, their common velocity after impact being represented by equation (701). If W, be exceedingly great as compared with W,,

438. Two ELASTIC BODIES IMPINGE UPON ONE ANOTHER IT 18 REQUIRED TO DETERMINE THE VELOCITY AFTER IMPACT.

If the impinging bodies be perfectly elastic, it is evident that after the period of their greatest compression is passed, they will, in the act of expanding their surfaces, exert mutual pressures upon one another, which are, in corresponding positions of the surfaces, precisely the same with those which they sustained whilst in the act of compression; whence it follows that the decrements of velocity experienced by that body whose motion is retarded by this expansion of the surfaces, and the increments acquired by that whose velocity is accelerated, will be equal to those before received in passing through corresponding positions, and therefore the whole decrements and increments thus received during the whole expansion equal to those received during the whole compression.

Now the velocity lost by W, during the compression ie represented by (V,-V); that lost by it during the expansion, or from the period of greatest compression to that when the bodies separate from one another, is therefore represented by the same quantity. But at the instant of greatest compression both bodies had the velocity V; the velocity v, of W, at the instant of separation is therefore V—(V1—V), or 2V-V,. In like manner, the velocity gained by W, during compression, and therefore during expansion, being represented by (VV), and its velocity at the instant of greatest compression by V, its velocity v, at the instant of separation is represented by V+(VFV2); or by 2VV, the sign being taken according as the

motion of the bodies before impact was in the same or opposite directions.

Substituting for V its value in these expressions (equation 701), and reducing, we obtain

If the bodies be perfectly elastic and equal in weight, v=V,,,V,; they therefore, in this case, interchange their velocities by impact; and if either was at rest before impact, the other will be at rest after impact.

29

If W, be exceedingly great as compared with W1, v,= -V, ±2V1, v=V,. In this case v, is negative, or the motion of the lesser body alters its direction after impact, when their motions before impact were in opposite directions; or when they were in the same direction, provided that 2V, be not greater than V1.

439. If the elasticities of the balls be imperfect, the force with which they tend to separate at any given point of the expansion is different from that at the corresponding point of the compression; the decrements and increments of the velocities, produced during given corresponding periods of the compression and expansion, are therefore different; whence it follows that the whole amounts of velocity, lost by the one and gained by the other during the two periods, are different: let them bear to one another the ratio of 1 to e. Now the velocity lost during compression by W, is under all circumstances represented by (V-V); that lost during expansion is therefore represented, in this case, by e (V, —V); therefore, v, V—e(V,―V)=(1+e)V-eV,. In like manner, the velocity gained by W, during compression is in all cases represented by (VV); that gained during expansion is therefore represented by e(VV); therefore, v=V+ e(VFV1)=(1+e) V FeV. Substituting for V, and reducing,

V1 = (W,—eW,)V, ±(1+e)W‚V, (707);

W1+W2
±(W‚—eW ̧)V2+(1+e)W ̧V,
W1+W2

....

(708).

440. IN THE IMPACT OF TWO ELASTIC BODIES, TO DETERMINE THE ACCUMULATED WORK, OR ONE HALF THE VIS VIVA, LOST BY THE ONE AND GAINED BY THE OTHER.

The vis viva lost by W, during the impact is evidently

represented by

{(1+ e)V— eV ̧ }

W

=

W

1

{(1.— e3)V,' + 2e(1 +e) VV, − (1 + e)'

W 1 (1 + e) (V. — V) { V, (1—e)+V(1+e)}.

Substituting in this expression its value for V (equation 701) reducing and representing by u, one half the vis viva lost by W, in its impact, or the amount by which its accumu lated work is diminished by the impact (Art. 67.),

Similarly, if u, be taken to represent one half the vis viva gained by W,, or the amount by which its accumulated work is increased by the impact, then

(1+e)W,W,(V, FV,) {2W,V,+(1—e)W ̧V‚±

2g(W1+W2)2
(1+e)W,V,}

(710).

441. Let u be taken to represent the whole amount of the work accumulated in the two bodies before their impact, which is lost during their impact. This amount of work is evidently equal to the difference between that gained by the one body and lost by the other; so that uu,-u,. Substituting the values of u, and u, from the preceding equations, and reducing, we obtain

u =

(1.

2

-e')W,W,(V‚¤V,)* . . . . (711).
2g(W1+W1)

This expression is equal to one half the vis viva lost during the impact of the bodies. If the bodies be perfectly elastic, e=1, and u=0. In this case there is no real loss of vis viva

in the impact, all that which the one body yields, during the impact, being taken up by the other.*

442. In the preceding propositions it has been supposed that the motions of the impinging body, and the body impinged upon, are opposed by no resistance whatever during the period of the impact. There is no practical case in which this condition obtains accurately. If, nevertheless, the resistance opposed to the motion of each body be small, Es compared with the pressure exerted by each upon the other, at any period of the impact, then it is evident that all the circumstances of the impact as it proceeds, and the motion of each body at the instant when it ceases, will be very nearly the same as though no resistance were opposed to the motion of either.t

443. As an illustration of the principle established in the last article, let it be required to determine the space through

It has been customary, nevertheless, to speak of a loss of vis viva in the impact of perfectly elastic bodies. This loss is in all such cases to be understood only as a loss experienced by one of the bodies, and not as an absolute loss. When the impinging bodies are perfectly elastic, it is evident that the one flies away with all the vis viva which is lost in the impact by the other. Let P1 and P2 represent resistances opposed to the motions of two imW1 W3 pinging bodies whose weights are W, and W.; also let g

fi, and fa re present the effective forces upon the two bodies at any period of the impact; then, by D'Alembert's principle,

or representing by t the time occupied in the impact, up to the period of greatest compression, by V their common velocity at that period, and by vi and their velocities at any period of the impact, and substituting for f, and f1⁄2 their values (equation 72),

Transposing and integrating between the limits 0 and t,

Now if P, and P, be not exceedingly great, the integral in the second member of the equation is exceedingly small as compared with its other terms, and may be neglected; the above equation will then become identical with equation (700).