COLLISION

, is the friction, percussion, or striking of bodies against one another.

Striking bodies are considered either as elastic, or non-elastic. They may also be either both in motion, or one of them in motion, and the other at rest.

When non-elastic bodies strike, they unite together as one mass; which, after collision, either remains at rest, or moves forward as one body. But when elastic bodies strike, they always separate after the stroke.

The principal theorems relating to the collision of bodies, are the following:

1. If any body impinge or act obliquely on a plane surface; the force or energy of the stroke, or action, is as the sine of the angle of incidence. Or the force | upon the surface, is to the same when acting perpendicularly, as the sine of incidence is to radius.

2. If one 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.

3. If the plane, acted on, be not absolutely fixed, it will move, after the stroke, in the direction perpendicular to its surface.

4. If a body A strike another body B, which is either at rest, or else in motion, either towards A or from it; 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.

Thus, first, if A with a momentum of 10, strike B at rest, and communicate to it a momentum of 4, in the direction AB. Then there will remain in A only a momentum of 6 in that direction: which together with the momentum of B, viz 4, makes up still the same momentum between them as before.—But 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 up 15, the same as 10 and 5, the motions before the stroke.—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 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.

5. If a hard and fixed plane be struck either by 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 with the same velocity with which it struck the plane.

6. The effect of the blow of the elastic body, upon the plane, is double to that of the non-elastic one; the velocity and mass being the same in both.

7. Hence, non-elastic bodies lose, by their collision, only half the motion that is lost by elastic bodies; the masses and velocities being equal.

8. If an elastic body A impinge upon 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 = CBE.

9. If the non-elastic body B, moving with the velocity V in the direction Bb, and the body b with the velocity v, strike each other, the direction of the mo- tion being in the line BC; then they will move after the stroke with a common velocity, which will be more or less according as, before the stroke, b moved towards B, or from B, or was at rest; and that common velocity, in each of these cases, will be as follows: viz, it will be (BV + bv)/(B + b) when b moved from B, (BV - bv)/(B + b) when b moved towards B, BV/(B + b) when b was at rest.

For example, if the bodies or weights, B and b, be 5lb and 3lb; and their velocities V and v, 60 feet and 40 feet per second; then 300 and 120 will be their momenta BV and bv, and the sum of the weights. Consequently the common velocity after the stroke, in the three cases above mentioned, will be thus, viz, in the first case, in the second case, 300/18 or 16 2/3 in the third case.

10. If two perfectly elastic bodies impinge on each other; their relative velocity is 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.

It is not meant however by this theorem, that each body will have the very same velocity after the impulse as it had before; but that the velocity of the one, after the stroke, will be as much increased, as that of the other is decreased, in one and the same direction. So, if the elastic body B move with the 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 only 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 again, and recede from each other after the stroke: in like manner, they would separate with the velocity V of B, if b were at rest before the stroke. Also the sum of the velocities of the one body, is equal to the sum of the others. But whether they move forwards or backwards 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 are as specified in the next theorem.

11. If the two elastic bodies B and b move directly towards each other, or directly from each other, the former with the velocity V, and the latter with the velocity v; then, after their meeting and impulse, the respective velocities of B and b in the direction BC, in the three cases of motion, will be as follow: viz, the velocity of B, the velocity of b, when the bodies both moved towards C before the stroke; and | the velocity of B, the velocity of b, when B moved towards C, and b towards B before the stroke; the velocity of B, the velocity of b, when b was at rest before the stroke.

12. The motions of bodies after impact, that ftrike each other obliquely, are thus determined.

Let the two bodies B, b, move in the oblique directions BA, bA, and strike each other at A with velocities which are in proportion to the lines BA, bA. Let CAH represent the plane in which the bodies touch in the point of concourse; to which draw the perpendiculars BC, bD, and complete the rectangles CE, DF. Now the motion in BA is resolved into the two BC, CA; and the motion in bA is resolved into the two bD, DA; of which the antecedents BC, bD are the velocities with which they irectly meet, and the consequents CA, DA are parallel, and therefore by these the bodies do not impinge on each other, and consequently the motions according to these directions will not be changed by the impulse; so that the velocities with which the bodies meet, are as BC or EA, and bD or FA. The motions therefore of the bodies B, b, directly striking each other with the celerities EA, FA, will be determined by art. 11 or 9, according as the bodies are elastic or non-elastic; which being done, let AG be the velocity, so determined, of one of them, as A; and since there remains also in the same body a force of moving in the direction parallel to BE, with a velocity as BE, make AH equal to BE, and complete the rectangle GH: then the two motions in AH and AG, or HI, are compounded into the diagonal AI, which therefore will be the path and celerity of the body B after the stroke. And after the same manner is the motion of the other body b determined after the impact.

13. The state of the common centre of gravity of bodies is not affected by the collision or other actions of those bodies on one another. That is, if it were at rest before their collision, so will it be also at rest after collision: and if it were moving in any direction, and with any velocity, before collision; it will do the very same after it.

See more upon this subject under the article PERCUSSION.