« PreviousContinue »
the motion of the body along the right line. For the same reason, the motions of all the other bodies, in their several directions, will still remain the same. Consequently their motions among themselves will continue the same, whether the including space be at rest, or be moved uniformly forward And therefore their mutual actions on one another, must also remain the same in both cases.
59. If a Hard and Fixed Plane be struck by either a Soft or a
Hard Unelastic Body, the Body will adhere to it. But if the Plane be struck by a Perfectly Elastic Body, it will rebound from it again with the same Velocity with which it struck the Plane.
For, since the parts which are struck, of the elastic body, suddenly yield and give way by the force of the blow, and as suddenly restore themselves again with a force equal to the force which impressed them, by the definition of elastic bodies ; the intensity of the action of that restoring force on the plane, will be equal to the force or momentum with which the body struck the plane. And, as action and re. action are equal and contrary, the plane will act with the same force on the body, and so cause it to rebound or move back again with the same velocity as it had before the stroke,
But hard or soft bodies, being devoid of elasticity, by the definition, having no restoring force to throw them off again, they must necessarily adhere to the plane struck.
60. Corol. 1. The effect of the blow of the elastic body, on the plane, is double to that of the unelastic one, the velocity and mass being equal in each.
For the force of the blow from the unelastic body, is as its mass and velocity, which is only destroyed by the resist. ance of the plane. But in the elastic body, that force is not only destroyed and sustained by the plane; but another also equal to it is sustained by the plane, in consequence of the restoring force, and by virtue of which the body is thrown back again with an equal velocity. And therefore the intensity of the blow is doubled.
61. Corol. 2. Hence unelastic bodies lose, by their collision, only half the motion lost by elastic bodies ; their mass and velocities being equal. -For the latter communicate double the motion of the former.
62. If an Elastic Body A impinge on a Firm Plane de at the
Point B, it will rebound from it in an Angle equal 10 thar in which it struck i'; or the Angle of Incidence will be equal to the Angle of Reflexion ; namely, the Angle ABD equal to the Angle Fbe.
LET AB express the force of the body a in the direction AB ;
с which let be resolved into the two AC, CB, parallel and perpendicular to the plane. -Take D Br and cf equal to ac, and draw B. Now action and reaction being equal, the plane will resist the direct force ce by another Bc equal to it, and in a contrary direction; whereas the other ac, being parallel to the plane, is not acted on or diminished by it, but still continues as before. The body is therefore reflected from the plane by two forces BC, BE, perpe..dicular and parallel to the plane, and therefore moves in the diagonal Bx by composition. But, because ac is equal to be or CF, and that bc is common, the iwo triangles BCA, BCF are mulually similar and equal; and consequently the angles at A and F are equal, as also their equal alternate angles ABD, FBE, which are the angles of incidence and reflexion.
63. To determine the Motion of Non-elastic Bodies when they strike each other Directly, or in the same Line of Direction,
Let the non-elastic body B, moving with the velocity v in the direction Bb, and the body 6 with B the velocity v, strike each other. Then, because the momentum of any moving body is as the mass into the velocity, BV = M is the momentum of the body B, and bu = m the momentum of the body b, which let be the less powerful of the two motions. Then, by prop. 10, the bodies will both move together as one mass in the direction bc after the stroke, whether before the stroke the body 6 moved towards c or towards B. Now, according as that motion of 6 was from or towards B, that is, whether the motions were in the same or contrary ways, the momentum after the stroke, in direction Bc, will
be the sum of difference of the momentums before the stroke ; namely, the momentum in direction Bc will be
BV + bu, if the bodies moved the same way, or
bv, if they moved contrary ways, and BV only, if the body 6 were at rest. Then divide each momentum by the common mass of matter B + 6, and the quotient will be the common velocity after the stroke in the direction Bc; namely, the common velocity will be, in the first case,
BV + bv
B + b
64. For example, if the bodies, or weights, B and b, be as 5 to 3, and their velocities v and v, as 6 to 4, or as 3 to 2, before the stroke ; then 15 and 6 will be as their momenrums, and 8 the sum of their weights; consequently, after the stroke, the common velocity will be as
15 + 6 21
=Zor 2 in the first case,
=cor lf in the second, and
or if in the third.
65. If two Perfectly Elastic Bodies impinge on one another :
their Relative Velocily will be the same boih Before and After the Impulse : that is, they will recede from each other with the same Velocity with which they approached and met.
For the compressing force is as the intensity of the stroke; which, in given bodies, is as the relative velocity with which they meet or strike. But perfectly elastic bodies restore themselves to their former 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 approach each other before the impulse. But the bodies are impelled from each other by this restoring force ; and therefore this torce, acting on the same bodies, will produce a relative velocity equal to that which they had before : or it will make the bodies rccede 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 bufore and after the impact.
66 Remark. It is not meant by this proposition, that each body will have the same velocity after the impulse as it had before ; for that will be varied according to the rela:ion of the masses of the two bodies ; but that the velocity of the one will be, after the stroke, as much increased as that of the other is decreased, in one and the same direction. So, if the elastic body B 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 from each other after the stroke. But if they meet each other, or the body b move contrary to the body B ; then they meet and strike with the velocity v + v, and it is with the same velocity that they separate and recede from each other after the stroke. But whether they move forward or backward afier the impulse, and with what particular velocities, are circumstances that depend on the various masses and velocities of the bodies before the stroke, and which make the subject of the next proposition.
67. To determine the Motions of Elastic Bodies after Striking
each other directly,
Let the elastic body B move in the direction BC, with the velocity B v; and let the velocity of the other body b be v in the same line ; which latter velocity v will be positive if 6 move the same way as B, but negative if b move in the opposite direction to B. Then their relative velocity in the direction Bc is v v; also the momenta before the stroke are bv and by, the sum of which is bv + bv in the direction Bc.
Again, put x for the velocity of B, and y for that of b, in the same direction BC, after the stroke ; then their relative velocity is y x, and the sum of their momenta Bx+by in the same direction.
But the momenta before and after the collision, estimated in the same direction, are equal, by prop. 10, as also the relative velocities, by the last prop. Whence arise these two equations :
viz Bv + bv = BX + by,
and V V = y
the velocity of B,
the velocity of b,
Btb both in the direction BC, when v and v are both positive, or the bodies both moved towards c before the collision. But if u be negative, or the body b moved in the contrary direction before collision, or towards B; then, changing the sign of v, the same theorems become (B-6) v - 2bv
, the velocity of B,
-, the veloc. of b, in the direction BC.
B +6 And if b were at rest before the impact, making its velocity D = 0, the same theorems give 6
2B B + 6
BI TV, the velocities in this case. And in this case, if the two bodies B and 6 be equal to
2B 2B each other ; then B b = o, and
B + b 2B give x = 0, and y = v; that is the body B will stand still, and the other body 6 will move on with the whole velocity of the former ; a thing which we sometimes see happen in playing at billiards; and which would happen much oftener if the balls were perfectly elastic.
68. If Bodies strike one another Obliquely, it is proposed to des
termine their Motions after the Stroke.
Let the two bodies B, b,
E move in the oblique directions BA, ba, and strike each other
E the impact. Let can repre
GH sent the plane in which the
I bodies touch in the point of
F concourse ; to which draw the perpendiculars Bc, bp, and complete the rectangles CE, DF. Then the motion in BA is re