14. A body is projected in a given direction, and with a given velocity, from a point in a plane making a given angle with the horizon; determine the distance of the body from the plane at any time (t), and the time which elapses before the body strikes the plane. 15. A body is projected from a point in an inclined plane, with a velocity V and at an angle a with the horizon; prove that, if ß be the angle of the plane, the range on the V2 {sin (2a-B) - sin ẞ}; and hence shew that the range is greatest when a = 45° + B 2 16. A body is projected with a given velocity along an inclined plane of given length; find the latus rectum of the parabola described by the body after leaving the plane. 17. Find the velocity and direction of projection of a ball, that it may be 100 feet above the earth at one mile distance, and may strike the ground at a distance of three miles. 1. IMPACT. The centres of two elastic balls M and M' move along the same straight line with velocities V and V' respectively. Find the velocity of each after impact, when 6M = 5M', V = 7 feet per second, 4V + 5 V' = 0, and modulus of elasticity = . 2. Two equilateral triangles are placed in the same vertical plane, and with their bases at a given distance from each other upon the same horizontal line: an inelastic body falls down the side of the first, moves along the space between the bases and up the side of the second triangle, the vertex of which it just reaches; given the side of the first triangle, find that of the second, and likewise the whole time of motion. 3. A perfectly elastic ball is let fall from a given point in the directrix of a parabola, the axis of which is vertical, and is reflected at the curve; determine the latus rectum of the parabola described. The position of a ball on a triangular billiard table being given; it is required to shew that there are three directions, in any one of which if the ball be struck, it will pursue the same course after being twice reflected at each side. The ball to be considered perfectly elastic. 5. PQ is a vertical line terminating in a hard horizontal plane at Q; a perfectly elastic ball being dropped from P meets another perfectly elastic ball rebounding with a known velocity from Q, and both are reflected back; find where they must meet in order that they may thus rebound from one another continually. 6. A and B are two balls of given elasticity; what must be the magnitude of a third ball, in order that the velocity communicated from A to B by the intervention of this ball may be equal to that communicated immediately from A to B? Determine also the limits within which the problem is possible. 7. Two balls are projected at the same instant from two given points in a horizontal plane, and in opposite directions, so as to describe the same parabola. What must be their relative magnitude, and their elasticity, in order that one of them may return through the same path as before, and the other descend vertically after impact? 8. A perfectly elastic body is projected from a point in a plane inclined at an angle a to the horizon; determine the angle at which it must be projected so that after striking the plane it may be reflected vertically upwards. 9. If the modulus of elasticity be, at what angle must a body be incident on a hard plane, that the angle between the directions before and after impact may be a right angle? 10. An inelastic body is projected from one angle along the side of a hexagon; and it moves in the interior of the hexagon, describing the different sides in succession; prove that the time of describing the first side time of describing the last :: 1 : 32. : 11. An imperfectly elastic body descending vertically from rest, meets a horizontal plane, which is moving uniformly in an opposite direction; given the distance between the body and the plane at first, and the modulus of elasticity, find the velocity of the plane, so that the body may return to the point from which it fell. 12. A number of elastic balls are placed in a right line. The first is made to start with a given velocity; determine the ratio of the balls, so that its momentum may be equally divided among the remainder. 13. If an elastic ball be projected at an angle 0 and with velocity V; prove that the sum of all the horizontal ranges 14. Two elastic balls A and B, (such that A = 3B) are placed on a horizontal table. A impinging on B at rest drives it perpendicularly against a hard vertical plane, and it meets A in returning at half its original distance. Find the modulus of elasticity. 15. P and Q are two weights connected by a string passing over a fixed pully, whereof P is the greater; at the end of t" an additional weight (9) is suddenly affixed to Q. Find the velocity of P at any assigned time. 16. A ball (elasticity e) is projected from a given point in the circumference of a circle: after being reflected twice at the circumference it returns to the point of projection. Required the direction of projection. 17. Prove that if a body be projected from one extremity of the diameter of a circle, in a direction making an angle with the diameter such that the body after one reflexion at the curve passes through the other extremity, then 18. If two perfectly elastic balls, the masses of which are in the ratio of 1: 3, meet directly with equal velocities, the larger one will remain at rest. MISCELLANEOUS PROBLEMS. 1. A body, projected in the direction of a uniform force, describes P and Q feet in the pth and 9th seconds respectively. Find the magnitude of the force and the velocity of projection. 2. Find the velocity acquired by an inelastic body descending through a system of three planes, the first being vertical, the second inclined at 45o, and the third at 15° to the horizon. 3. Find the elasticity of two bodies A and B, and their proportion to each other, so that when A impinges upon B at rest, ▲ may remain at rest after impact, and B move on with an nth part of A's velocity. 4. Two weights are connected by a string which passes through a hole in a horizontal plane, one rests upon the plane, the other falls under the action of gravity; determine the motion. 5. Uniform force is defined as that which generates equal velocities in equal times; would it be correct to define it as that which generates equal velocities while the body moves through equal spaces? 6. A rocket ascending vertically, with an initial velocity of 100 feet per second, explodes when at its greatest height; the interval between the sound of the explosion reaching the place of starting and a place a quarter of a mile distant is 1 second. Determine the velocity of sound. 7. An inelastic body moving along the interior of a regular polygon, will describe all the sides uniformly, if it commences moving uniformly; but the velocities with which it describes the successive sides will decrease according to a geometrical progression. Prove this, and in the case of a hexagon find the ratio of the velocities with which the first and last sides are described. We subjoin here a few illustrations of the formula for the time of oscillation of a pendulum. If the length of the pendulum bel, and the bob be made to describe a cycloidal arc, or the arc of vibration be so small that the difference between it and a cycloidal arc may be neglected, the formula for the time of a semi-vibration is T-TV g It will be assumed, that above the earth's surface the force of gravity varies inversely as the square of the distance from the earth's centre; so that if g' be the value of gravity at a small height h above the earth's surface, and R the earth's radius, Also it will be assumed, that within the earth the force of gravity varies directly as the distance from the centre; so that if g" be the value of gravity at a small depth d below the earth's surface, |