Page:The American Cyclopædia (1879) Volume XI.djvu/331

 MECHANICS 319 104* FIG. 3. _ m+m, -- The following are among the propositions of Huygens on elastic bodies. If a body meets an equal body at rest, after im- pact the former will be at rest, but the latter will acquire the velocity of the impelling body. If two equal bodies moving with unequal velo- cities strike one another, they will after impact move with interchanged velocities. Any body, however large, is moved by any body, how- ever small, and moving with any velocity what- soever. If two bodies meet from opposite directions whose velocities are inversely in proportion to their quantity, each will rebound with the velocity with which it approached. These propositions may be experimentally veri- fied by the apparatus shown in fig. 3. Let two elastic balls be suspended at equal heights by slender threads of a radius corresponding to the graduated arc al. If one of the balls is raised and allowed to fall against the other, this will be impelled to a corresponding height ; if both are raised to the same height and let fall, they will re- bound to their original heights (theoretically) ; and if one is let fall from a greater height than the other, they will rebound with reciprocal velocities. If several balls are hung between them so that they touch each other, and im- pact is made by a terminal ball, all the in- termediate balls will remain at rest, the im- pulse being transmitted through them to the opposite ball. If a perfectly elastic ball is thrown obliquely against a smooth plane, it will be reflected so as to make the angle of reflection equal to the angle of incidence. If the ball is imperfectly elastic, the angle of re- flection will be greater than the angle of in- cidence ; and if the ball is perfectly inelastic, it will not be reflected, but will slide upon the plane. Uniformly accelerated Rectilinear Motion; Laws of Falling Bodies. As a con- sequence of the property of inertia, when a body has been put in motion and all force is removed, it tends to continue in motion with uniform velocity and in a right line. But if the force which caused the motion is uniform and constant, the body will receive equal in- crements of force during equal spaces of time, and therefore its motion will become uniformly accelerated. The most uniform constant known force at the surface of the earth is gravitation, and it is by its means that the laws of uni- formly accelerated motion are studied. A body falling through the air does not in fact have its motion uniformly accelerated, because of the resistance of the air ; but it can be proved by experiment as well as by a process of reasoning that such would be its motion in a vacuum. That all bodies near the surface 542 VOL. xi. 21 58 of the earth tend to fall with equal velocities without regard to their density or bulk, is shown by the common guinea and feather ex- periment in a tube exhausted of air. The element of resistance of the air will not there- fore, in considering the subject, be taken into account. The velocity of a falling body, in consequence of gravity being a uniform and constant force, will be in proportion to its time of fall, and its average velocity during any given space of time will be at the middle of that space ; and therefore the velocity which a body in falling acquires at the end of any period of time will be double the aver- age velocity from the commencement. Let the figures 1, 2, 3, 4, 5, 6 on the left of the column in the adjoining diagram represent the number of seconds during which a body falls from rest ; they will also represent the velocities acquired at the ends of the seconds. Now, as the aver- age velocity during the first two seconds is acquired at the end of the first second, and as the aver- age velocity of the next two sec- onds is acquired at the end of the third second, if we represent the space fallen through during the first two seconds by S, the space fallen through during the third and fourth seconds will be represented by 3 S, and for simi- lar reasons the space fallen through during the fifth and sixth seconds will be represent- ed by 5 S. Therefore, during equal successive portions of time a body falls from rest through successive spaces represented by the odd num- bers 1, 3, 5, 7, &c. ; so that if the space through which it falls in one second be called a unit, that through which it falls during the first two seconds will be 1 + 3=4, and that through which it falls from rest in three seconds will be 1 + 3 + 5=9 spaces; or the spaces through which a body falls from rest during 1, 2, 3, 4, &c., seconds will be proportional to the squares of these numbers. We thus by a process of reasoning, and without the assistance of ex- perimental demonstration, arrive at the follow- ing laws of falling bodies : 1. The velocity acquired by a body in falling is proportional to the time of fall. 2. The spaces through which a body falls in equal successive periods of time vary as the odd numbers 1, 3, 5, &c. 3. The whole space through which a body falls from rest is proportional to the square of the time. 4. The velocity acquired by a falling body during any period of time, if continued uni- formly, will carry it through twice the space in the same time ; a law which follows from the second law, by which the spaces fallen through during equal successive periods of time increase by a constant quantity, which is twice the space through which a body falls from rest during one second. This constant quantity, which in this latitude is 32'16 ft., is usually.