latera recta of the parabolas, which they severally describe, lie on a cone, of which the axis is vertical, and the vertical angle 2 tan 1 2.

18. Two heavy particles are projected from two given points in the same vertical line with equal velocities in parallel directions; prove that tangents, drawn to the path of the lower, will cut off, from the path of the upper, arcs described in equal times.

19. A heavy particle is projected from a given point with given velocity so as to pass through another given point; prove that, in general, there will be two parabolic paths which the particle may describe: and give a geometrical construction to determine their foci. Also find the locus of the second point in order that there may be only one parabolic path.

20. A barrel of a rifle sighted to hit the centre of the bull's eye, which is at the same height as the muzzle and distant a yards from it, would be inclined at an angle a to the horizon. Prove that, if the rifle be wrongly sighted so that the elevation is a +0,0 being small compared with a, the target will be hit at O above the centre of the bull's eye.

a height a

cos 2a

cos2 α

21. Particles are projected in the same plane and from the same point, so that the parabolas described are equal; prove that the locus of the vertices of all these parabolas is an equal parabola.

22. If a particle be projected obliquely up an inclined plane in the form of a rectangle of given sides, one side being horizontal, find the velocity such that it is projected from one corner and leaves the plane horizontally at the other: and shew that the ratio of the horizontal range after leaving the plane to that described on the plane is the sine of the angle of elevation of the plane.

23. If a particle projected from a given point with velocity V in a direction making an angle a with the horizon meet a second particle let fall from the directrix at the instant of projection,

shew that the distance of the line described by the latter particle

from the point of projection of the former is

cot a.

2g

IMPACT.

1. 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+5V'=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.

4. 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. 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?

7. A perfectly elastic ball 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.

8. If the modulus of elasticity be, at what angle must a ball be incident on a hard plane, that the angle between the directions before and after impact may be a right angle?

9. An inelastic ball 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.

:

10. 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.

11. If an elastic ball be projected at an angle and with velocity V; prove that the sum of all the horizontal ranges

:

=

V2 sin 20

12. 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.

13. If two perfectly elastic balls, the masses of which are

G. P.

7

in the ratio of 1:3, meet directly with equal velocities, the larger one will remain at rest.

14. Prove that if a ball of elasticity e, be projected from one extremity of the diameter of a horizontal circle, in a direction making an angle & with the diameter such that the ball after one reflexion at the curve passes through the other extremity, then

15. A ball of elasticity e is projected obliquely up an inclined plane so that the point of impact at the third time of striking the plane is in the same horizontal line with the point of projection; prove that the distances from this line of the points of first and second impact, are in the ratio 1 : e.

16. A series of perfectly elastic balls are arranged in the same straight line, one of them impinges upon the next, and so on; prove that, if their masses form a geometrical progression of which the common ratio is 2, their velocities after impact will form a geometrical progression of which the common ratio

17. The intersection of two inclined planes is horizontal, and a perfectly elastic ball, projected from a point on one plane, strikes the other, the inclinations of the planes to the horizon being the halves of the inclinations to the vertical of the directions of projection and first impact respectively; determine the whole subsequent motion.

18. From what height must a perfectly elastic ball be let fall into a hemispherical bowl, in order that it may rebound horizontally at the first impact, and strike the lowest point of the bowl at the second.

MISCELLANEOUS PROBLEMS.

1. A particle projected in the direction of a uniform acceleration, describes P and Q feet in the pth and 9th seconds respectively. Find the magnitude of the acceleration and the velocity of projection.

2.

Find the velocity acquired by an inelastic particle descending through a system of three planes, the first being vertical, the second inclined at 45°, and the third at 15° to the horizon.

3. Find the elasticity of two particles A and B, and their proportion to each other, so that when A impinges upon B at rest, A may remain at rest after impact, and B move on with an nth part of A's velocity.

4. Uniform acceleration 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?

5. 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.

6. If two parabolas be placed with their axes vertical, vertices downwards, and foci coincident, prove that there are three chords down which the time of descent of a particle under the action of gravity from one curve to another is a minimum, that one of these is the principal diameter, and the other two make an angle of 60° with it on either side.

7. Two perfectly elastic balls A and B impinge upon each other. First A impinges upon B at rest, and goes off in a