CHAPTER III. RECTILINEAR MOTION. 74. THE simplest case of motion which we have to consider is that of a particle in a straight line. This may be due to a force acting at every instant in the direction of motion; or the particle may be supposed to be constrained to move in a straight line by its being enclosed in a straight tube of indefinitely small bore. As already mentioned, § 64, we shall in every case suppose the mass of the particle to be unity. 75. A particle moves in a straight line, under the action of any forces, whose resultant is in that line; to determine the motion. Let P be the position of the particle at any time t, f the resultant acceleration acting always along OP, O being a fixed point in the line of motion. Let OP=x, then the equation of motion is d2x In this equation ƒ may be given as a function of x, of dx dt or of t, or of any two or all three combined; but in any case the first and second integrals of the equation (if they can be dx obtained) will give and x in terms of t; that is, the position and velocity of the particle at any instant will be known. The only one of these cases which we will now consider is that in which fƒ is given as a function of x; those in which f is a function of dx dx or of dt and x, being reserved for the Chapter on Motion in a Resisting Medium: while those in which f involves t explicitly possess little interest, as they cannot be procured except by special adaptations; and can even then appear only in an incomplete statement of the circumstances of the particular arrangement. The simplest supposition we can make is that ƒ is constant. 76. A particle, projected from a given point with a given velocity, is acted on by a constant force in the line of its motion; to determine the position and velocity of the particle at any time. Let A be the initial position of the particle, Pits position at any time t, v its velocity at that time, and f the constant acceleration of its velocity. Take any fixed point 0 in the line of motion as origin, and let OA=a, OP=x. The equation of motion is C being a constant to be determined by the initial circumstances of the motion. Suppose the particle projected from A in the positive direction with velocity V, then when t=0, v=V; hence C = V, and But when t=0, x = a; hence C' = a, and Equations (2) and (3) give the velocity and position of the particle in terms of t; and the velocity may be determined in terms of by eliminating t between them: but the same result will be obtained more directly by multiplying (1) by and integrating. This gives the equation of energy dx dt 77. The most important case of the motion of a particle under the action of a constant force in its line of motion is that in which the force is gravity. For the weights of bodies in the same latitude at small distances above the Earth's surface may be considered constant, and therefore if we denote the kinetic measure of the earth's attraction by g, and consider the particle to be projected vertically downwards; equations (2), (3), (4) of § 76 become x being measured as before from a fixed point O in the line of motion. As a particular instance suppose the particle to be dropped from rest at 0. At that instant A coincides with O, and a=0, V = 0. The last of these equations may also be obtained from 78. As another particular instance, suppose the particle to be projected vertically upwards. Here it must be remembered that if we measure x upwards from the point of projection, the force tends to diminish x and the equation of motion is In other respects the solution is the same. Taking, therefore, a = 0 in equations (A) and changing the sign of g, we obtain V From equation (4) we see that the velocity continually diminishes, and becomes zero when t; and from (6) that the height corresponding to v=0, or the greatest height to which the particle will ascend, is After this the velocity V2 2g becomes negative, or the particle begins to descend, and (5) shews that it will return to the point of projection when 2 V t == g as x then becomes 0; and the velocity with which it returns to that point is, by (6), equal to the velocity of projection. 79. A particle descends a smooth_inclined plane_under the action of gravity, the motion taking place in a vertical plane perpendicular to the intersection of the inclined with any horizontal plane; to determine the motion. Let P be the position of the particle at any time t on the inclined plane OA, OP=x its distance from a fixed point 0 in the line of motion, and let a be the inclination of OA to the horizontal line AB. The only impressed force on the particle is its weight g which acts vertically downwards, and this may be resolved into two, g sin a along, and g cosa perpendicular to, OA. Besides these there is the unknown force R, or the reaction of the plane, which is perpendicular to OA: but neither this nor the component g cos a can affect the motion along the plane. The equation of motion is therefore the solution of which, as g sin a is constant, is included in that of the proposition of § 77, and all the results for particles moving vertically under the action of gravity will be made to apply to it by writing g sin a for g. Thus, if the particle |