11. A sphere sliding on a smooth horizontal plane impinges in succession on two smooth vertical planes at right angles to each other; prove that the velocity of the sphere after the second impact is parallel to its velocity before the first impact. 12. Two spheres moving in parallel directions with equal and opposite mass-velocities impinge; prove that after impact they will move in parallel directions with velocities in the inverse proportion of their masses, 13. A smooth small sphere of elasticity e slides down a smooth inclined plane of height h and inclination a, and impinging on a smooth horizontal plane at the foot of the first describes a parabola; find the range. 14. A particle of elasticity e is projected from a horizontal plane so as to impinge on a given vertical wall at a distance a ft., and after one rebound on the horizontal plane returns to the point of projection; the velocity of projection being u velos, find the dire tion of projection. 15. A sphere of elasticity e is projected at an angle a from the middle of the floor of a rectangular room h ft. high, ab ft. wide, so that after impinging on a wall, on the ceiling and on the opposi e wall, it returns to the point of projection; find equations to deterrine the velocity of projection. 16. A smooth inelastic particle slides along the sides of a regular polygon of n sides under the action of no forces; prove that its velocity after passing m angular points is diminished in the ratio of 2 TT m COS ) to i. 17. Two masses m lbs. and m' lbs., moving with u velos and u velos respectively impinge, so that u, u make angles a, á with the line of impact; prove that if after impact they are moving with v velos and v velos respectively, then mm' mv2 + m' v'?=mu+ m'u". (is - e) (u cosa - u' cosa')?. m+m' NEWTON'S LAWS OF MOTION. 134. Newton's Laws of Motion are LEX I. Corpus omne perseverare in statu quo quiescendi vel movendi uniformiter in directum, nisi quatenus illud a viribus impressis cogitur statum suum mutare. LEX II. Mutationem motus proportionalem esse vi motrici impressa, et fieri secundum lineam rectam qua vis illa imprimitur. LEX III. Actioni contrariam semper et æqualem esse reactionem : sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi. They may be translated as follows: I. Every body perseveres in its state of rest or of moving uniformly in a straight line, except in so far as it is made to change that state by external forces. II. The change of the mass-velocity of a body is numerically equal to the impulse which produces it and is in the same direction. III. To every action there corresponds an equal and opposite reaction; that is to say, the actions of two bodies upon each other are always equal and in opposite directions. Newton's Laws, I and II, follow immediately from Art. 31, provided, we define Impulse as the equivalent of a Force acting continuously for a definite interval; for force produces acceleration; therefore impulse produces additional velocity. We may therefore in Art. 33 replace the words force and acceleration by the words impulse and velocity; and then Art. 33 is a statement of what is implied by Laws I and II. Conversely, Art. 31 follows immediately from Laws I and II, provided we define force as, that which acting continuously for a definite interval is equivalent to an impulse; for impulse is that which produces additional velocity; therefore force must be that which produces acceleration. Example. Consider a mass of 10 tons on a smooth horizontal plane (for instance a railway carriage on smooth horizontal rails), to which an horizontal impulse is applied by means of a hammer. Suppose the mass of the hammer to be 2 lbs. and the velocity 32 velos; and suppose the hammer just brought to rest by the impact. The impulse communicates the mass velocity 64 pound-velos to the 10 tons. When the 10 tons is in motion, by making the hammer impinge again with a velocity 32 velos greater than the velocity of the 10 tons and bringing it relatively to rest by the impact, we can arrange that the impulse 64 pound-velos can be repeated any number of times. Now suppose men employed to apply to the 10 tons in succession a series of impulses at intervals of Too of a second; each impulse communicating 64 pound-velos to the 10 tons, and suppose that they apply 100 impulses per second. In such a case, if the force of each impulse lasts exactly too of a second, and is uniform, then the 10 tons is acted on by a continuous force. 64 pound-velos are communicated to the 10 tons every tio of a second ; that is, 6400 pound-velos per second, and The motion is that produced by a force of 6400 poundals; or by the weight of about 200 lbs. 135. Let us suppose that there is such a thing as an instantaneous impulse. Suppose an impulse w pulses applied to a particle of unit mass (in the direction of the motion of the particle) at the beginning of each of a series of n intervals, each of a seconds, such that nt= t. The particle will receive at the beginning of each interval an additional velocity w. 8 L. D. Next suppose the intervals diminished, their number increased, and the ratio of the impulse to the interval kept the same so that Then in the limit, when the number n is infinitely increased, the motion of the particle is that of the uniformly increasing velocity of a celos which is caused by continuous force. 136. Hence an impulse which is equivalent to a force continuing for some given interval may be looked upon as a series of infinitely small impulses applied at infinitely small intervals whose sum is equal to the given interval. The magnitude of a continuous force is measured by the rate at which it produces impulse. 137. It is not necessary however in the above to suppose the impulse instantaneous. Let each impulse produce the velocity w velos gradually and uniformly in the interval 7 secs.; then, whether the intervals 7 secs. are large or small, the motion of the particle is uniformly accelerated motion such as, by Art. 31, is produced by continuous force. 138. But just as the properties of curves are studied by supposing them to be derived from polygons, in which the sides are made to become infinitely numerous and infinitely short; so it is sometimes convenient to imagine our continuous force acting for a certain interval to be equivalent to a series of small impulses; these impulses being instantaneous, and becoming, in the limit, infinitely quick and infinitely small. a 139. PROP. To prove the formula s= ut + fat of Art. 21, by Newton's method. Suppose a point to be moving in a straight line so that its velocity receives at the beginning of each one of a series of equal intervals of 7 secs. an additional velocity w velos. Let n of these intervals of 7 secs. make up t secs. Let u be the velocity of the point at the beginning of the interval t seconds. The velocity during the rth interval 7 seconds is (u +rw) velos. The distance passed over in the course of that interval is (u +rw) ft. So that the whole distance passed over in t seconds is {(u +w) + +(u + 2w) + + ... + (u + nw) T} feet - (7 that is in{2u1 + (n + 1)w} feet that is {ut + 1nwt + {wt} feet. [Compare Art. 24.] Now suppose the interval 7 seconds be diminished and consequently the number n increased so that nt still equals t. Suppose also that the rate at which velocity is added is kept unaltered, so that is a constant number = a; and T Then the distance passed over in t seconds is ať {ut n This is true however great n may be; and hence it is true when n is greater than any assigned number; in which case the distance passed over in t seconds is ut + lat. 140. The following proposition is an instructive example of Newton's method. |