The last of these equations may also be obtained from 83. 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 acceleration 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 From equation (4) we see that the velocity continually V diminishes, and becomes zero when t=—; and from (6) that g the height corresponding to v = 0, or the greatest height to which the particle will ascend, is After this the velocity 2g* becomes negative, or the particle begins to descend, and (5) shews that it will return to the point of projection when 2V = g , as a then becomes 0; and the velocity with which it returns to that point is, by (6), equal to the velocity of projection. 84. A particle descends a smooth inclined plane under the acceleration 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 O 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 cos a perpendicular to OA. Besides these there is the unknown force R, the pressure on 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 sina is constant, is included in that of the proposition of § 82, 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 start from rest at 0, we get from equations (1), (2), (3) of § 82 by this means, 85. Equation (3) proves an important property with regard to the velocity acquired at any point of the descent. For, draw PN parallel to AB, and let it meet the vertical line through O in Ñ, then if v be the velocity at P, we have Comparing this with equation (3) of § 82, we see that the velocity at P is the same as that which a particle would acquire by falling freely from rest through the vertical distance ON; in other words the velocity at any point, of a particle sliding down a smooth inclined plane, is that due to the vertical height through which it has descended; a particular case of the conservation of energy. 86. Again from (2) we derive immediately the following curious and useful result. The times of descent down all chords drawn through the highest or lowest point of a vertical circle are equal. Let AB be the vertical diameter of the circle, AC any chord through A; join BC; then if T be the time of descent down AC, we have by equation (2) of § 84, which, being independent of the position of the chord, gives the same time of descent for all. A B It may similarly be shewn that the time of descent down all chords through B is the same. In fact parallel chords, drawn through A and B respectively, are of equal length. To find the straight line of swiftest descent to a given curve from any point in the same vertical plane, all that is required is to draw a circle having the given point as the upper extremity of its vertical diameter, and the smallest which can meet the curve. Hence if BC be the curve, A the point, draw AD vertical; and, with centre in AD, describe a circle passing through A and touching BC. Let P be the point of contact, then AP is the required line. T. D. For, if we take any 5 other point, p, in BC, Ap cuts the circle in some point q, and time down Ap> time down Aq, i.e. > time down AP. If the given curve be not plane, or if it be required to find the straight line of swiftest descent to a surface, a sphere must be described passing through A, with centre in AD, so as to touch the curve or surface; and the proof is precisely as before. 87. In § 84 we have supposed the inclined plane to be smooth, but the motion will still be constantly accelerated when the plane is rough. For since there is no motion, and therefore no acceleration, perpendicular to OA (see fig. § 84), we must have 0=R-g cos a. (§ 69). If u then be the coefficient of kinetic friction, which is μ known by experiment to be independent of the velocity of the particle, the retardation of friction will be uR or μg cos a, and the equation of motion will become. the second member still being constant, and the solution therefore similar to those we have already considered. 88. When a particle moves under an attraction in its line of motion, varying directly as the distance of the particle from a fixed point in that line, to determine the motion. Let O be the fixed point, P the position of the particle at any time t, v its velocity at that time, and let OP=x. Then if μ be the acceleration of a particle due to the attraction at a unit of distance from O, which is supposed known, and is called the strength of the attraction, the acceleration at P will be ux, and if it be directed towards O will tend to diminish. Therefore the equation of motion is |