surface. For it is in that direction that the force is exertūd. 136. Prop. If one body A, strike another body B, which is either at rest or moving towards the body A, or moving from it, but with a less velocity than that of A; then the momenta, or quantities of motion, of the two bodies, estimated in any one direction, will be the very same after the stroke that they were before it. For, because action and re-action are always equal, and in contrary directions, art. 20, whatever momentum the one body gains one way by the stroke, the other must just lose as much in the same direction ; and therefore the quantity of motion in that direction, resulting from the motions of both the bodies, remains still the same as it was before the stroke. * 137. Thus, if A with a momentum of 10, strike B at rest, and communi- (3–O—H cate to it a momentum of 4, in the A B C direction AB. Then A will have only a momentum of 6 in that direction; which, together with the momentum of B, viz. 4, make up still the same momentum between them as before, namely 10. 138. If B were in motion before the stroke, with a momentum of 5, in the same direction, and receive from A an additional momentum of 2. Then the motion of A after the stroke will be 8, and that of B, 7 ; which between them make 15, the same as 10 and 5, the motions before the stroke. 139. Lastly, if the bodies move in opposite directions, and meet one another, namely, A with a motion of 10, and B, of 5; and A communicate to B a motion of 6 in the direction AB of its motion. Then, before the stroke, the whole motion from both, in the direction of AB, is 10–5 or 5. But, after the stroke, the motion of A is 4 in the direction AB, and the motion of B is 6–5 or 1 in the same direction AB ; 'therefore the sum 4 + 1, or 5, is still the same motion from both, as it was before. 140. PRop. The motion of bodies included in a given space, is the same with regard to each other, whether that space be at rest, or more uniformly in a right line. For, if any force be equally impressed both on the body and the line on which it moves, this will cause no change in the motion of the body along the right line. For the same reason, the motions of all the other bodies, in their several *irections, will still remain the same. Consequently their *motions armong 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. 141. PRop. 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. 142. 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 resistance 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. 143. 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. 144. PRop. If an elastic body A impinge on a firm plane DE at the point B, it will rebound from it in an angle equal to that in which it struck it; or the angle of incidence will be equal to the angle of reflection; namely, the angle Abd equal to the angle F.B.E. Let AB express the force of the body A in the direction AB; which let be resolved into the two Ac, CB, parallel and perpendicular to theplane.—Take BE and CF equal to AC, and draw Br. Now action and re-action being equal, the plane will resist the direct force ch by another be 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 lic, BE, perpendicular and pa. rallel to the plane, and therefore moves in the diagonal Bf by composition. But, because Ac is equal to Be or cf, and that BC is common, the two triangles hea, Bcf are mutually similar and equal ; and consequently the angles at A and F are equal, as also their equal alternate angles ABD, FRE, which are the angles of incidence and reflection. 145. Prop. 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 di. $—?—a rection Bb, and the body b with B the velocity v, strike each other. Then, because the momentum of any moving body is as the mass into the velocity, by = M is the momentum of the body B, and br = m the momentum of the body b, which let be the less powerful of the two motions. Then, by art. 136, the bodies will both move together as one mass in the direction no after the stroke, whether before the stroke the body b moved towards c or towards B. Now, according as that motion of b was from or towards B, that is, whether the motions were in the same or contrary ways, the momentum after the stroke, in direction ac, will be the sum or difference of the momentums before the stroke ; namely, the momentum in direction Bc will be Bv + be, if the bodies moved the same way, or - Bv — be, if they moved contrary ways, and bv only, if the body b were at rest. Then divide each momentum by the common mass of matter B + b, and the quotient will be the common velocity after the stroke in the direction Bo ; namely, the common velocity will be, in the first case, av-le in the 2d w-o, and in the 3d-to-. H-i-b B-H. b Corol. v — "..."- . X b, the veloc. lost by B. 146. 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 momen tums, and 8 the sum of their weights ; consequently, after the stroke, the common velocity will be as 147. Prop. If two perfectly elastic bodies impinge on one another, their relative velocity will be the same both 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 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. 148. 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 relation of the masses of the two bodies ; but that the velocity of the one will be, after the stroke, so much increased, and the other decreased, as to have the same difference as before, 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-Hv, 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 after 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.—It may further be remarked, that the sums of the two velocities, of each body, before and after the stroke, are equal to each other. Thus, v, v' being the velocities before the impact, if z and y be the corresponding ones after it; since v — v = y – r, therefore v + r = v + y. 149. Prop. 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 § ; C v; and let the velocity of the other body b be v in the same line ; which latter velocity v will be positive if b 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 bu, the sum of which is by + bi, in the direction BC. Again, put a 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 – r, and the sum of their momenta Br + by in the same direction. But the momenta before and after the collision estimated in the same direction, are equal, by art. 136, as also the relative velocities, by the last prop. Whence arise these two equations: viz. BV -- bu = Ba + by, and v — v = y — a ; the resolution of which equations gives _ (B-b}v4-2hp B+b , the velocity of B, - –(e–b)n+2Ev y = — B+b , the velocity of b. B+ So that the velocity lost b 25 o that the velocity los y - is i;(x-1), 2 and the velocity gained by b is o —v); which two velocities are in the ratio of b to B, or reciprocally as the two bodies themselves. Corol. 1. The velocity lost by B drawn into B, and the velocity gained by b drawn into b, give each of them |