A View of Sir Isaac Newton's Philosophy

by Henry Pemberton

4. After this manner is determined in every case the result of a body in motion striking against another at rest. The same principles will also determine the effects, when both bodies are in motion.

5. Let two equal bodies move against each other with equal swiftness. Then the force, with which each of them presses forwards, being equal when they strike; each pressing in its own direction with the same energy, neither shall surmount the other, but both be stopt, if they be not elastic: for if they be elastic, they shall from thence recover new motion, and recede from each other, as swiftly as they met, if they be perfectly elastic; but more slowly, if less so. In the same manner, if two bodies of unequal bigness strike against each other, and their velocities be so related, that the velocity of the lesser body shall exceed the velocity of the greater in the same proportion, as the greater body exceeds the lesser (for instance, if one body contains twice the solid matter as the other, and moves but half as fast) two such bodies will entirely suppress each other’s motion, and remain from the time of their meeting fixed; if, as before, they are not elastic: but, if they are so in the highest degree, they shall recede again, each with the same velocity, wherewith they met. For this elastic power, as in the preceding case, shall renew their motion, and pressing equally upon both, shall give the same motion to both; that is, shall cause the velocity, which the lesser body receives, to bear the same proportion to the velocity, which the greater receives, as the greater body bears to the lesser: so that the velocities shall bear the same proportion to each other after the stroke, as before. Therefore if the bodies, by being perfectly elastic, have the sum of their velocities after the stroke equal to the sum of their velocities before the stroke, each body after the stroke will receive its first velocity. And the same proportion will hold likewise between the velocities, wherewith they go off, though they are elastic but in a less degree; only then the velocity of each will be less in proportion to the defect of elasticity.

6. If the velocities, wherewith the bodies meet, are not in the proportion here supposed; but if one of the bodies, as A, has a swifter velocity in comparison to the velocity of the other; then the effect of this excess of velocity in the body A must be joined to the effect now mentioned, after the manner of this following example. Let A be twice as great as B, and move with the same swiftness as B. Here A moves with twice that degree of swiftness, which would answer to the forementioned proportion. For A being double to B, if it moved but with half the swiftness, wherewith B advances, it has been just now shewn, that the two bodies upon meeting would stop, if they were not elastic; and if they were elastic, that they would each recoil, so as to cause A to return with half the velocity, wherewith B would return. But it is evident from hence, that B by encountring A will annul half its velocity, if the bodies be not elastic; and the future motion of the bodies will be the same, as if A had advanced against B at rest with half the velocity here assigned to it. If the bodies be elastic, the velocity of A and B after the stroke may be thus discovered. As the two bodies advance against each other, the velocity, with which they meet, is made up of the velocities of both bodies added together. After the stroke their elasticity will separate them again. The degree of elasticity will determine what proportion the velocity, wherewith they separate, must bear to that, wherewith they meet. Divide this velocity, with which the bodies separate into two parts, that one of the parts bear to the other the same proportion, as the body A bears to B; and ascribe the lesser part to the greater body A, and the greater part of the velocity to the lesser body B. Then take the part ascribed to A from the common velocity, which A and B would have had after the stroke, if they had not been elastic; and add the part ascribed to B to the same common velocity. By this means the true velocities of A and B after the stroke will be made known.

7. If the bodies are perfectly elastic, the great Huygens has laid down this rule for finding their motion after concourse[46]. Any straight line C D (in fig. 4, 5.) being drawn, let it be divided in E, that C E bear the same proportion to E D, as the swiftness of A bore to the swiftness of B before the stroke. Let the same line C D be also divided in F, that C F bear the same proportion to F D, as the body B bears to the body A. Then F G being taken equal to F E, if the point G falls within the line C D, both the bodies shall recoil after the stroke, and the velocity, wherewith the body A shall return, will bear the same proportion to the velocity, wherewith B shall return, as G C bears to G D; but if the point G falls without the line C D, then the bodies after their concourse shall both proceed to move the same way, and the velocity of A shall bear to the velocity of B the same proportion, that G C bears to G D, as before.

8. If the body B had stood still, and received the impulse of the other body A upon it; the effect has been already explained in the case, when the bodies are not elastic. And when they are elastic, the result of their collision is found by combining the effect of the elasticity with the other effect, in the same manner as in the last case.

9. When the bodies are perfectly elastic, the rule of Huygens[47] here is to divide the line C D (fig. 6.) in E as before, and to take E G equal to E D. And by these points thus found, the motion of each body after the stroke is determined, as before.

10. In the next place, suppose the bodies A and B were both moving the same way, but A with a swifter motion, so as to overtake B, and strike against it. The effect of the percussion or stroke, when the bodies are not elastic, is discovered by finding the common motion, which the two bodies would have after the stroke, if B were at rest, and A were to advance against it with a velocity equal to the excess of the present velocity of A above the velocity of B; and by adding to this common velocity thus found the velocity of B.

11. If the bodies are elastic, the effect of the elasticity is to be united with this other, as in the former cases.

12. When the bodies are perfectly elastic, the rule of Huygens[48] in this case is to prolong C D (fig. 7.) and to take in it thus prolonged C E in the same proportion to E D, as the greater velocity of A bears to the lesser velocity of B; after which F G being taken equal to F E, the velocities of the two bodies after the stroke will be determined, as in the two preceding cases.