(35) If a particle fall to a center of force (D); determine the constant force which would produce the effect in the same time, and compare the final velocities. (36) Find the equation to the curve down each of whose tangents a particle will slide to the horizontal axis in a given time. (37) A sphere is composed of an infinite number of free particles, equally distributed, which gravitate to each other without interfering; supposing the particles to have no initial velocity, prove that the mean density about a given particle will vary inversely as the cube of its distance from the center. CHAPTER IV. PARABOLIC MOTION. 100. In this chapter we intend to treat principally of the motion of a free particle which is subject to the action of forces whose resultant is parallel to a given fixed line. The simplest case of course will be when that resultant is constant. The problem then becomes the determination of the motion of a projectile in vacuo, since the attraction of the earth may be considered within moderate limits as constant and parallel to a fixed line. This we will now consider. 101. A free particle moves under the action of a vertical force whose magnitude is constant; to determine the form of the path, and the circumstances of its description. Taking the axis of x horizontal and in the vertical plane and sense of projection, and that of y vertically upwards, it is evident that the particle will continue to move in the plane of xy, as it is projected in it, and is subject to no force which would tend to withdraw it from that plane. The equations of motion then are if g be the kinetic measure of the force. Suppose that the point from which the particle is projected is taken as origin, that the velocity of projection is V, and that the direction of projection makes an angle a with the axis of x. The first and second integrals of the above equations will then be These equations give the co-ordinates of the particle and its velocity parallel to either axis for any assumed value of the time. Eliminating t between equations (2) we obtain the equation to the trajectory, viz. which shews that the particle will move in a parabola whose axis is vertical, and vertex upwards. By comparing this with the equation to a parabola, we find for the co-ordinates x, y, of the vertex Now if v be the velocity of the particle at any point of its path, To acquire this velocity in falling, from rest, the par ticle must have fallen, § 77, through a height V2 2g -y, i.e. through the distance from the directrix. 2g 103. To find the time of flight along a horizontal plane. Put y = 0 in equation (3). of x are 0 and sin a cos a. 212 The corresponding values But the horizontal velocity 2 V sin a is V cosa. Hence the time of flight is g ; and, ceteris paribus, varies as the sine of the inclination to the horizon of the direction of projection. 104. To find the time of flight along an inclined plane passing through the point of projection. Let its intersection with the plane of projection make an angle B with the horizon; it is evident that we have only to eliminate y between (3) and y = x tan B. This gives for the abscissa of the point where the projectile meets the plane, 105. To find the direction of projection which gives the greatest range on a given plane. The range on the horizontal plane is sin 2a. For a given value of V this will be greatest when That this may be a maximum for a given value of V we must equate to zero its differential coefficient with respect to a, which gives the equation cos a cos (a — ẞ) — sin a sin (a — B) = 0, or cos (2a) = 0; Hence the direction of projection required for the greatest range makes with the vertical an angle that is, it bisects the angle between the vertical and the plane on which the range is measured. 106. To find the elevation necessary to the particle's passing through a given point. Suppose the point in the axis of x and distant a from the origin. Then we must have Let a' be the smallest positive angle whose sine is may be pro see there are two directions in which a particle jected so as to reach the given point, and that these are equally inclined to the direction of projection (a = (a Π which gives the greatest range. |