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A body is said to be perfectly hard, when its component par. cannot be separated, or moved among themselves, by any finite force.
A soft vody consists of particles which give way upon the applica tion of the least force or impression. Such a body does not again, of itself, resume its figure, but remains altered.
An elastic body has a natural tendency to recover its former figure, after it has been altered by compression. By a perfectly elastic body, we mean one which recovers its figure with a force equal to that which was employed in compressing it. Almost all bodies, with which we are acquainted, are elastic in a greater or less degree, but none perfectly so. In steel balls, the force of elasticity is to the compressing force as 5 to 9, and, in glass balls, as 15 to 16.
74. The impact of two bodies is said to be direct, when their centres of gravity move in the right line which passes through the point of impact.
On the Impact of hard, or non-elastic Bodies. 75. If two bodies have no elasticity, when they conie into contact by direct impact, there is no force whatever to separate thein again ; consequently they must either remain at rest, or move on uniformly together.
76. To determine the several particulars relating to the motion, velocities, and direction of two percutient bodies, let A and B, fig. 10, represent the bodies or quantities of matter, V and v their velocities, then will AV be the momentum of A, and Bv that of B.
77. Then, if the body A strikes the body B in motion, and both move the same way, or towards the same part, as from A towards B, then the sum of their motions, in that direction, will be AV + Bv, and the velocity of both bodies after the stroke, towards the same part,
AV+Bv will be
=U. For the velocity is always as the momentum divided by the mass (27).
78. If one of the bodies, as B, has a contrary direction, in which case the bodies will meet, then the momentum of B will have a negative sign, viz. -Bv ; and the sum of the motion towards the same part, will be AV-Bv; and the velocity, after the collision, will be AV-Bv A+B
79. The sum of the motions towards the same parts, is the sam. before and after the stroke. For let them both move the same way, and let A strike B ; then, by that impulse, the motion of B, viz. Bu, will be augmented, and become Bv+x after the stroke ; but, because action and re-action are equal (20). The body B will re-act upon A, and produce an equal effect by the impact; that is, it will diminish the motion of A by the same quantity f, so that its motion after the stroke will be 'AV-X; but the sum of the motions of both, after the stroke, is AV-:+Bu+r=AV+Bv, the
sum of the motions before the stroke. And the same may be shown of the bodies meet.
80. The magnitude of the stroke will be proportional to the quantity I, because that is the whole effect or mutation produced in the motion of each body. The greatness of the stroke is, therefore, measured by the loss x, which the body, having the greatest momentum, sustains in its motion.
81. In the above theurem (77), if the body B be supposed at rest, then v=0, and Bu vanishes ; the velocity, then, after the stroke,
AV is A+B = U; and, consequently, AV = (A + B)U. Whence
A+B U: V:; A:AB, and V=
A In the same theorem, if we suppose the bodies equal, viz. A=B ; then, if the bodies tend the same way, the velocity, after the stroke, will be V+v=U; or V-v=U, if they meet.
87. Cor. 1. If the bodies are equal, and one of them, as B, at rest, then AV
U; or the velocity, after the stroke, is equal to half that of the A+B striking body,
83. Cor. 2. If B, at rest, exceed A infinitely in magnitude, then because AV is infinitely small in respect of A+B, therefore so is U in respect of V (81); consequently, U will vanish, or the body A impinging against any firm, immoveable object, will, after the stroke, be at rest.
84. Cor. 3. If equal bodies, moving with equal velocities, meet, they will mutually destroy each other's motions ; for in this case, AV-Bv=o, and,
AV-Bv consequently, A+B
=U=0; or both bodies remain at rest after the
85. The momentum of the body A, after the stroke, is AU= A'V +ABv
A'V+ABv_ABV + ABv (by art. 77); therefore AVA+B
A+B AB =AFB(VFv)= the loss of motion in the body A after the struke. But A+B
is a constant quantity ; therefore the loss of motion in A is as VFv; and, consequently, the magnitude of the stroke in bodies, tending the same way, is as V-V; and as V+v, if they meet. And if B be at rest, then v=0, and the stroke will be as V, the velocity of the percutient body.
B 86. Cor. The velocity lost by A==
A(A+B)(V FO) ATBIV Fv); and since B gains the same quantity of motion that A loses, the velocity gained
ATB(VFv.) Hence the velocity lost by A, and that gained by B, are reciprocally as the bodies themselves.
Er. Let the weights of two bodies, A and B, be 10 and 6, and beir velocities 12 and 8 respectively : then, when they move the same
A way, the velocity gained by B will be A+B(V—v)=16x4=2}.
B Again, the velocity lost by A=
A+B(V—v)=fóx4=11. When they move in opposite directions, we shall find that the relo city gained by B, in the direction of A's motion, equal 12į; and the velocity lost by A equal 7; and the velocity of both bodies, in the same direction, will be found 4.
ON THE COLLISION OF ELASTIC BODIES. 87. If two perfectly elastic bodies infringe on one another, their relative velocity will be the same both before and after the impulse; or, in other words, they will recede from each other with the same velocity as that with which they approached or met. For the compressing force is as the intensity of the stroke, and this (85) is as the relative velocity with which they meet or strike. But perfectly elastic bodies restore themselves to their foriner figure, by the same force by which they were compressed ; that is, the restoring force is equal to the compressing force, or to the force with which the bodies approached each before the impulse. But the bodies are impelled from each other by this restoring force; and, therefore, this force acting on the same bodies, will produce a relative velocity equal to that which they had before; or, it will make the bodies recede from each other with the same velocity with which they before approached, or so as to be equally distant from one another, at equal times, before and after the impact.
We do not here mean to say, that each body will have the same velocity after the inpact as it had before, but that the velocity of the one will, after the stroke, be so much increased, and that of the other so much decreased, as to have the same difference as before, in one and the same direction. So that if the elastic body A, fig. 10, move with a velocity V, and overtake the elastic body B, moving the same way with the velocity v; then their relative velocity, or that with which they strike, is V-V, and it is with this same velocity that they separate after the stroke. But if they meet each other, or the body B move contrary to the body A, then they meet and strike with the velocity V+ v, and it is with this same velocity that they separate and recede from each other after the stroke.
It may further be observed, that the sums of the two velocities of each body, before and after the stroke, are equal to each other. Thus. if V and v be the velocities before impact, and x and y the correse donding one after it, since V-v=y—, we also have, by transposi. tion, V+x=vty.
88. Let the elastic body A, fig. 11, move in the direction AC, with the velocity V; and let the velocity of the other body B, in the same line, be v; which latter velocity v will be positive, if B move the sanje way as A, but negative if B move in the opposite direction to A.
Then their relative velocity in the direction AC is V-V; also the momenta before the stroke are AV and Bv, the sum of which is AV + Bo a the direction AC.
Again, put x for the velocity of A, and y for that of B, in the same direction AC after the stroke ; then their relative velocity is yand the sum of their momenta, in the same direction, is Ar+ By.
But the momenta before and after the collision, estimated in the same direction, are equal, whatever be the nature of the bodies (by 79), as are also their relative velocities, in the case of elastic bodies, by the preceding article. Whence arise these two equations,
AV + Buz Ax+By,
and V-v=y-*. The resolution of these equations gives (A-BV+2Bv
for the velocity of A,
the velocity of B. From the above values of x and y, we find V-t, or the velocng 2B
2A = BIV—v) and y—v, or that gained by B=A and these velocities are in the ratio of B to A, or, reciprocally, as the bodies themselves. 89. Cur. The velocity lost by A drawn into A, and the velocity gained by B
24B drawn into B, give each of them
A+B(V—o) for the momentum gained by the one and lost by the other in consequence of the stroke; this increment and decrement being equal, they cancel one another, and leave the same momentum AV + Bv after the impact, as it was before it.
90. Hence, also, AV? +Bv=Aro + By, or the sum of the vires viviarum, or living forces, are the same both before and after impact.
For since AV+Bv=Ax+By, by transposition,
we find AV-Ar=By-Bv and (87) V+r=y+v, these two equations multiplied together give AV-Ax'=By-Bv, or
AV + Bv=B.: +By. 91. Cor. If u be negative, or if the body B move in the contrary directions betore collision, or towards A, by changing the sign of v, the same theorems become,
the velocity of A,
the velocity of B in the direction AC.)
A+B 92. If B were at rest before the impact, we have v=0; and, consequently, A-B
2A Ā+B*V, and y=A+B *V, the velocities in this case.
Again, suppose B at rest, and the two bodies, A and B, equal to
2A each other; then A-B=0, and XV= XV=V. There
2A fore, the body A will stand still, and the body B will move forward with the whole velocity of the former. This we sometimes see happen in playing at billiards, and it would occur much more frequently, it the balis were perfectly elastic. 93. Cor. In the expressions
A+B Let A=B, then <=+, and y=V, and therefore the bodies interchange veloc. ties.
94. If the bodies be elastic only in a partial degree, the sum of the momenta will still be the same, both before and after collision, but the velocities after, will be less than in the case of perfect elas. ticity, in the ratio of the imperfection. Hence, employing the same notation as before, the two equations of article 88, will become
AV + Bv=Ar+By,
and V-v= *(7—5); where m to n denotes the ratio of perfect to imperfect elasticity. The resolution of these two equations gives the following values of s and y, viz.
mtn B x=V
A+B m+n? A
y=ut for the velocities of the two bodies, after impact, in the case of imperfect elasticity : and these equations agree with the former when n=m.
From these expressions for x and y, we may easily solve the principal problems relating to the collision of imperfectly elastic bodies.
95. When a perfectly hard body impinges obliquely on a perfectly hard and immoveable plane AB, fig. 12, in the direction CD, after impact it will move along the plane, and the velocity before impact : the velocity after :: radius : cos. < CDA.
Take CD to represent the motion of the body before impact ; draw CE parallel, and DE perpendicular, to AB. The motion CD may be resolved into the two CE, ED, art. 65, of which ED is wholly employed in carrying the body in a direction perpendicular to the plane, and since the plane is immoveable, this motion will be wholly destroyed. The other motion CE, which is employed in carrying the body parallel to the plane, will not be effected by the impact ; and, consequently, there being no force to separate the body and the plane, the body will move along the plane ; and it will describe DB=CE,