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and therefore

Of course we have also

Loss

v − U = e (V − v).

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U' — v = e (v − V').

These results may be put in words thus:-The relative velocity of either of the bodies with regard to the centre of inertia of the two is, after the completion of the impact, reversed in direction, and diminished in the ratio e : 1.

301. Hence the loss of kinetic energy, being, according to §§ 267, 280, due only to change of kinetic energy relative to the centre of inertia, is to this part of the whole as 1 - e2: 1.

Thus

V

− v)3 + ¿M' (v − V')3.

Initial kinetic energy = } (M + M') v2 + ↓ M (
= { (M + M') v2 + }M (v − U)2 + { M' (U' — v)2.
= { (1 − e2) {M (V − v)2 + M' (v − V')'}.

Final

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Direct impact of

spheres.

tion of

impact.

302. When two elastic bodies, the two balls supposed above Distribufor instance, impinge, some portion of their previous kinetic energy after energy will always remain in them as vibrations. A portion of the loss of energy (miscalled the effect of imperfect elasticity) is necessarily due to this cause in every real case.

Later, in our chapter on Properties of Matter, it will be shown as a result of experiment, that forces of elasticity are, to a very close degree of accuracy, simply proportional to the strains (§ 154), within the limits of elasticity, in elastic solids which, like metals, glass, etc., bear but small deformations without permanent change. Hence when two such bodies come into collision, sometimes with greater and sometimes with less mutual velocity, but with all other circumstances similar, the velocities of all particles of either body, at corresponding times of the impacts, will be always in the same proportion. Hence the velocity of separation of the centres of inertia after impact Newton's will bear a constant proportion to the previous velocity of tai law con approach; which agrees with the Newtonian Law. It is there- perfect fore probable that a very sensible portion, if not the whole, of the loss of energy in the visible motions of two elastic bodies, after impact, experimented on by Newton, may have been due

sistent with

elasticity.

tion of

Distribu- to vibrations; but unless some other cause also was largely energy after operative, it is difficult to see how the loss was so much greater with iron balls than with glass.

impact.

303. In certain definite extreme cases, imaginable although not realizable, no energy will be spent in vibrations, and the two bodies will separate, each moving simply as a rigid body, and having in this simple motion the whole energy of work done on it by elastic force during the collision. For instance, let the two bodies be cylinders, or prismatic bars with flat ends, of the same kind of substance, and of equal and similar transverse sections; and let this substance have the property of compressibility with perfect elasticity, in the direction of the length of the bar, and of absolute resistance to change in every transverse dimension. Before impact, let the two bodies be placed with their lengths in one line, and their transverse sections (if not circular) similarly situated, and let one or both be set in motion in this line. The result, as regards the motions of the two bodies after the collision, will be sensibly the same if they are of any real ordinary elastic solid material, provided the greatest transverse diameter of each is very small in comparison with its length. Then, if the lengths of the two be equal, they will separate after impact with the same relative velocity as that with which they approached, and neither will retain any vibratory motion after the end of the collision.

304. If the two bars are of unequal length, the shorter will, after the impact, be exactly in the same state as if it had struck another of its own length, and it therefore will move as a rigid body after the collision. But the other will, along with a motion of its centre of gravity, calculable from the principle that its whole momentum must (§ 267) be changed by an amount equal exactly to the momentum gained or lost by the first, have also a vibratory motion, of which the whole kinetic and potential energy will make up the deficiency of energy which we shall presently calculate in the motions of the centres of inertia. For simplicity, let the longer body be supposed to be at rest before the collision. Then the shorter on striking it will be left at rest; this being clearly the result in the case of

tion of

impact.

e = 1 in the preceding formulæ (§ 300) applied to the impact Distribuof one body striking another of equal mass previously at rest. energy after The longer bar will move away with the same momentum, and therefore with less velocity of its centre of inertia, and less kinetic energy of this motion, than the other body had before impact, in the ratio of the smaller to the greater mass. It will also have a very remarkable vibratory motion, which, when its length is more than double of that of the other, will consist of a wave running backwards and forwards through its length, and causing the motion of its ends, and, in fact, of every particle of it, to take place by "fits and starts," not continuously. The full analysis of these circumstances, though very simple, must be reserved until we are especially occupied with waves, and the kinetics of elastic solids. It is sufficient at present to remark, that the motions of the centres of inertia of the two bodies after impact, whatever they may have been previously, are given by the preceding formula with for e the value

M'

M'

where M' and M are the smaller and the larger mass respectively.

305. The mathematical theory of the vibrations of solid elastic spheres has not yet been worked out; and its application to the case of the vibrations produced by impact presents considerable difficulty. Experiment, however, renders it certain, that but a small part of the whole kinetic energy of the previous motions can remain in the form of vibrations after the impact of two equal spheres of glass or of ivory. This is proved, for instance, by the common observation, that one of them remains nearly motionless after striking the other previously at rest; since, the velocity of the common centre of inertia of the two being necessarily unchanged by the impact, we infer that the second ball acquires a velocity nearly equal to that which the first had before striking it. But it is to be expected that unequal balls of the same substance coming into collision will, by impact, convert a very sensible proportion of the kinetic energy of their previous motions into energy of vibrations; and generally, that the same will be the case when equal or unequal masses of different substances come into colli

Distribu

sion; although for one particular proportion of their diameters, energy after depending on their densities and elastic qualities, this effect will

tion of

impact.

be a minimum, and possibly not much more sensible than it is
when the substances are the same and the diameters equal.

Moment of an impact about an axis.

Ballistic pendulum.

306. It need scarcely be said that in such cases of impact as that of the tongue of a bell, or of a clock-hammer striking its bell (or spiral spring as in the American clocks), or of pianoforte hammers striking the strings, or of a drum struck with the proper implement, a large part of the kinetic energy of the blow is spent in generating vibrations.

307. The Moment of an impact about any axis is derived from the line and amount of the impact in the same way as the moment of a velocity or force is determined from the line and amount of the velocity or force, §§ 235, 236. If a body is struck, the change of its moment of momentum about any axis is equal to the moment of the impact round that axis. But, without considering the measure of the impact, we see (§ 267) that the moment of momentum round any axis, lost by one body in striking another, is, as in every case of mutual action, equal to that gained by the other.

Thus, to recur to the ballistic pendulum-the line of motion of the bullet at impact may be in any direction whatever, but the only part which is effective is the component in a plane perpendicular to the axis. We may therefore, for simplicity, consider the motion to be in a line perpendicular to the axis, though not necessarily horizontal. Let m be the mass of the bullet, v its velocity, and p the distance of its line of motion from the axis. Let M be the mass of the pendulum with the bullet lodged in it, and k its radius of gyration. Then if w be the angular velocity of the pendulum when the impact is complete,

mvp = Mk3w,

from which the solution of the question is easily determined.

For the kinetic energy after impact is changed (§ 241) into its equivalent in potential energy when the pendulum reaches its position of greatest deflection. Let this be given by the angle 0 then the height to which the centre of inertia is raised is h (1- cos 0) if h be its distance from the axis. Thus

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In

an expression for the chord of the angle of deflection.
practice the chord of the angle is measured by means of a
light tape or cord attached to a point of the pendulum, and
slipping with small friction through a clip fixed close to the posi-
tion occupied by that point when the pendulum hangs at rest.

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308. Work done by an impact is, in general, the product of Work done the impact into half the sum of the initial and final velocities of the point at which it is applied, resolved in the direction of the impact. In the case of direct impact, such as that treated in § 300, the initial kinetic energy of the body is MV2, the final MU2, and therefore the gain, by the impact, is

}M (U2 – V2),

or, which is the same,

But M(U – V) is (§ 295) equal to the amount of the impact.
Hence the proposition: the extension of which to the most
general circumstances is easily seen.

M (U – V) . § (U + V).

Let be the amount of the impulse up to time 7, and I the

whole amount, up to the end, T. Thus,—

+= ['Pdr, I= ["Pdr; also P = dr

L=

Whatever may be the conditions to which the body struck is subjected, the change of velocity in the point struck is proportional to the amount of the impulse up to any part of its whole time, so that, if M be a constant depending on the masses and conditions of constraint involved, and if U, v, V denote the component velocities of the point struck, in the direction of the impulse, at the beginning, at the time 7, and at the end, respectively, we have

2.Wel

v=U+

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J
Hence, for the rate of the doing of work by the force P, at the
instant t, we have

V = U +

I

ગુમ

P
Pv=PU +
M'

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Ballistic pendulum.

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