The velocity of the centre of gravity estimated in the direction at right angles to the line of impact, by Art. 126, is mu sin a+m'u' sin a' before impact, and m+m mv sin ẞ+m'r' sin B' after impact; and these are equal by Art. 114. Thus the component velocity of the centre of gravity in two directions is the same after impact as before; and therefore the resultant velocity is the same in magnitude and direction after impact as before. 130. It follows from the investigation of Art. 126, that if two bodies move in straight lines, each with uniform velocity, then their centre of gravity moves also in some straight line, with uniform velocity. Hence we may establish the following proposition: the centre of gravity of two projectiles, which are moving simultaneously, describes a parabola. For suppose at any instant that gravity ceased to act; then each body would move in a straight line with uniform velocity, and so would also the centre of gravity. The effect of gravity in a given time is to draw each body down a vertical space which is the same for each body, and which varies as the square of the time; and the centre of gravity is drawn down through the same vertical space. Hence, by reasoning as in Art. 51, we find that the path of the centre of gravity is a parabola. 131. By the method of Arts. 126 and 127, we may establish the following result: If f and f' be the accelerations, estimated in any direction, of two moving bodies, whose masses are m and m' respectively, the acceleration of the centre of gravity of the two bodies estimated in the mf+m'f same direction is m+m' And this result may be extended to the case of any number of bodies: see Art. 128. 1. A body weighing 4 lbs., and another weighing 8 lbs. are moving in the same direction, the former with the velocity of 8 feet per second, and the latter with the velocity of 2 feet per second: determine the velocity of the centre of gravity. 2. Equal bodies start from the same point in directions at right angles to each other, one with the velocity of 4 feet per second, and the other with the velocity of 5 feet per second: determine the velocity of the centre of gravity. 3. In the system of Art. 89 supposing the initial velocity zero, find the velocity of the centre of gravity at the end of a given time. 4. A heavy body hanging vertically draws another along a smooth horizontal plane; supposing the initial velocity zero, find the horizontal and the vertical velocity of the centre of gravity at any instant. 5. Shew that the centre of gravity in the preceding Example describes a straight line with uniform acceleration. 6. In the system of Art. 92 supposing the initial velocity zero, find the velocity of the centre of gravity at the end of a given time resolved parallel to the two planes. 7. Shew that the centre of gravity in the preceding Example describes a straight line with uniform acceleration. 8. Two balls are dropped from two points not in the same vertical line, and strike against a horizontal plane, the elasticity being perfect: shew that the centre of gravity of the balls will never re-ascend to its original height, unless the initial heights of the balls are in the ratio of two square numbers. 9. Three equal particles are projected, each from one angular point of a triangle along the sides taken in order, with velocities proportional to the sides along which they move: shew that the velocity of the centre of gravity estimated parallel to each side is zero; and hence that the centre of gravity remains at rest. 10. P, Q, R are points in the sides BC, CA, AB reBP CQ AR spectively of the triangle ABC, such that shew that the centre of gravity of the coincides with that of the triangle ABC. = CP AQ BR` triangle PQR XII. Laws of Motion. General Remarks. 132. We propose in the present Chapter to make some general remarks concerning the Laws of Motion. It is not necessary that a student should devote much attention to this Chapter on his first reading of the subject. He should notice the points which are here considered, and when in his subsequent course he finds any difficulty as to these points he can examine the remarks which bear upon the difficulty. 133. We will here repeat the Laws of Motion. I. Every body continues in a state of rest or of uniform motion in a straight line, except in so far as it may be compelled to change that state by force acting on it. II. Change of motion is proportional to the acting force, and takes place in the direction of the straight line in which the force acts. III. To every action there is always an equal and contrary reaction: or the mutual actions of any two bodies are always equal and oppositely directed in the same straight line. It is manifest that instead of Laws of Motion it would be more accurate to call these statements, Laws relating to the connexion of force with motion. 134. We have already observed that the motion of a body here considered is of that kind in which all the points of the body describe curves identical in form, though varying in position. For example, when we speak of the motion of a falling body we mean such a motion that every point of the body describes a straight line. The motion which is here considered is called motion of translation, to distinguish it from motion of rotation, which we do not consider. 135. We have also stated, in connexion with the distinction just explained, that the Laws of Motion ought to be enunciated with reference to particles rather than to bodies. It might appear to a beginner that there can be little advantage in studying the theory of the motion of particles, because in practice we are always concerned with bodies of finite size. But it is not difficult to shew the importance and value of a sound theory of the motion of particles. For it is easy to conceive that a solid body is made up of particles, and that the forces acting may be such as to make the motion of one particle exactly the same as the motion of another; and so the motion of the body is known when that of one particle is known. The case of a falling body illustrates this remark; see also Art. 81. Again, it is shewn in the higher parts of Mechanics that the motion of the centre of gravity of a rigid body is exactly the same as the motion of a particle having a mass equal to the mass of the rigid body, and acted on by forces equal and parallel to those which act on the rigid body. Although the student could not at the present stage follow the reasoning by which this remarkable result is obtained, nor even fully apprehend the result itself, yet he may readily perceive that great interest is thus attached to the theory of the motion of particles. 136. Up to the end of the sixth Chapter we considered the effect which a force produces on the velocity of a body without regard to the mass of the body moved. It is usual to apply the name accelerating force to force so considered; and hence the two following definitions are used: Force considered only with respect to the velocity generated is called accelerating force. Force considered with respect to the mass to which velocity is communicated as well as to the velocity generated is called moving force. The terms tend to confuse a beginner, because they lead him to suppose that there are two kinds of force. There is really only one kind of force, namely, that which is called moving force in the foregoing definitions; for when force acts it always acts on some body. It is not necessary to make any use of the term accelerating force: when the beginner hears or reads of an accelerating force ƒ he must remember that this means a force which produces the acceleration ƒ in the motion of the body which is considered. 137. We have followed Newton in our enunciation of the Laws of Motion; but it is necessary to observe that this course is not universally adopted. Many writers in effect divide Newton's Second Law into two, which they term the Second and Third Laws, presenting them thus: Second Law. When forces act on a body in motion each force communicates the same velocity to the body as if it acted singly on the body at rest. Third Law. When force acts on a body the momentum generated in a unit of time is proportional to the force. Then Newton's Third Law is presented as another principle which must be admitted to be true, although apparently not difficult enough or not important enough to be ranked formally with the Laws of Motion. We have followed Newton for two reasons. In the first place, his mode of stating the Laws of Motion seems, to say the least, as good as any other which has been proposed; and in the second place, there is very great advantage in a uniformity among teachers and students as to the first principles of the subject, and this uniformity is more likely to be secured under the authority of Newton than under that of inferior names. 138. We have given in Art. 47 Newton's form of the parallelogram of velocities; some writers omit this, and supply its place by a purely geometrical proposition, which is substantially as follows: |