49. Corol. 5. If three forces be in equilibrio by their mutual actions; the line of direction of each force, as DC, passes through the opposite angle c of the parallelogram formed by the directions of the other two forces. 30. Remark. These properties, in this proposition and its corollaries, hold true of all similar forces whatever, whether they be instantaneous or continual, or whether they act by percussion, drawing, pushing, pressing, or weighing; and are of the utmost importance in mechanics and the doctrine of forces. ON THE COLLISION OF BODIES. PROPOSITION IX. 51. If a Body strike or act Obliquely on a Plain Surface, the Force or Energy of the Stroke, or Action, is as the Sine of the Angle of Incidence. Or, the Force on the Surface is to the same if it had acted Perpendicularly, as the Sine of Incidence is to Radius. AG LET AB express the direction and the absolute quantity of the oblique force on the plane DE; or let a given body A, moving with a certain velocity, impinge on the plane at B; then its force will be to the action on the plane, as radius to the sine of the angle ABD, or as AB, to AD or BC, drawing AD and BC perpendicular, and ac parallel to DE. D B E For, by prob. 7, the force AB is equivalent to the two forces AC, CB; of which the former AC does not act on the plane, because it is parallel to it. The plane is therefore only acted on by the direct force CB, which is to AB, as the sine of the angle BAC, or ABD, to radius. 52. Corol. 1. If a body act on another, in any direction, and by any kind of force, the action of that force on the second body, is made only in a direction perpendicular to the surface on which it acts. For the force in AB acts on DE only by the force CB, and in that direction. 53. Corol. 2. If the plane DE be not absolutely fixed, it will move, after the stroke, in the direction perpendicular to its surface. For it is in that direction that the force is exerted. VOL. II. R PROPOSITION PROPOSITION X. 54. 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 &; 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, 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. 55. Thus, if A with a momentum of 10, strike в at rest, and communicate to it a momentum of 4, in the direction AB. Then A will have only A B c a momentum of 6 in that direction; which, together with the momentum of в, viz. 4, make up still the same momentum between them as before, namely, 10. 56. If в 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. 57 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 в 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. PROPOSITION XI. 58. The Motion of Bodies included in a Given Space, is the same with regard to each other, whether that Space be at Rest, or move uniformly in a Right Line. FOR, if any force be equally impressed both on the body and the line in which it moves, this will cause no change in the 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. PROPOSITION XII. 59. Ifa 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 reaction 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 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. 61. Cor. 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. PROPOSITION PROPOSITION XIII. 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 to that in which it struck it ; 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; which let be resolved into the two AC, CB, parallel and perpendicular to the plane.-Take BE and CF equal to ac, and draw BF. Now action and reaction being equal, the plane will resist the direct force cв 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, perpendicular and parallel to the plane, and therefore moves in the diagonal Br by composition. But, because AC is equal to BE or CF, and that BC is common, the two triangles BCA, BCF are mutually similar and equal; and consequently the angies at a and F are equal, as also their equal alternate angles ABD, FBE, which are the angles of incidence and reflexion. PROPOSITION XIV. 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 вb, and the body b with the velocity v, strike each other. Then, because the momentum of any moving body is as the mass into the velocity, BVM is the momentum of the body в, and bum 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 b moved towards c or towards B. Now, according as that motion of b was from or towards в, that is, whether the motions were in the same or contrary ways, the momentum after the stroke, in direction BC, will be be the sum or 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 bv, if they moved contrary ways, and By only, if the body b were at rest. Then divide each momentum by the common mass of matter Bb, 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, 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 momentums, and 8 the sum of their weights; consequently, after the stroke, the common velocity will be as 65. 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 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 cach other with the bodies are same |