9. A body is projected along a smooth horizontal table with a velocity g: find the length to which the table must be prolonged in the direction of the body's motion, so that the body after leaving the table may strike a point whose distances measured horizontally and vertically from the point of projection are 3g and 2g respectively.

10. A heavy particle is projected from a given point in a given direction so as to touch a given straight line: give a geometrical construction for determining the point of contact and the elements of the path described. If the direction of projection be not fixed, find the path so that the velocity of projection may be the least possible.

11. A chord is drawn joining any point on the circumference of a vertical circle with the lowest point: shew that if a heavy body slide down this chord the parabola which it describes on leaving the chord has its directrix passing through the upper end of the chord.

12. Chords are drawn joining any point on the circumference of a vertical circle with the highest and lowest points; a heavy body slides down the lower chord: shew that the parabola which it will describe after leaving the chord is touched by the other chord, and that the locus of the points of contact is a circle.

13. A heavy body is projected from one fixed point so as to pass through another which is not in the same horizontal line with it: shew that the locus of the focus of its path is an hyperbola.

14. A force acting uniformly during one tenth of a second produces in a given body the velocity of one mile per minute: compare the force with the weight of the body.

15. One end of a string is fastened to a weight P; the string passes over a fixed pully, and under a moveable pully, and has its other end attached to a fixed point; weight Q is attached to the moveable pully: determine the motion, supposing the three portions of the string all parallel.

T

16. In the formulæ of Art. 101 shew that if the velocities u and u are each increased by the same quantity, so are the velocities v and v'.

17. From the formulæ of Art. 101 determine the values of u' and v' if m=em'; also if m'=em.

18. A body of given mass is moving in a given direction: determine the magnitude and the direction of a blow which will cause it to move with the same velocity in a direction at right angles to the former.

19. A projectile at the instant it is moving with the velocity u at an inclination a to the horizon impinges on a vertical plane which makes an angle ẞ with the plane of motion of the projectile: find the velocity after impact.

20. Small equal spherical balls of perfect elasticity are placed at the corners of a regular hexagon; one of them is projected with the velocity u, so as to strike all the others in succession and to pass through its original position: find the velocity with which it returns.

21. In the preceding Example shew that each of the five balls starts at right angles to an adjacent side of the hexagon; and find the velocity with which each starts.

22. Two perfectly elastic balls of equal mass impinge; shew that if the directions of motion after impact are parallel, the cosine of the angle between their original directions is equal to the ratio of the product of the velocities after impact to the product before impact.

23. Of two equal and perfectly elastic balls one is projected so as to describe a parabola, and the other is dropped from the directrix so as just to fall upon the first when at its highest point: determine the position of the vertex f the new parabola.

24. A mark in a vertical wall appears elevated at an le B at a certain point in a horizontal plane; from this point a ball is projected at the mark and after striking it returns to the point of projection: shew that if a be the angle of projection tan a = (1+e) tan ß.

T. M.

22

25. A plane is inclined at an angle ẞ to the horizon; a particle is projected from a point in the plane at an inclination a to the horizon, with the velocity u, and the particle rebounds from the plane: find the time of describing n parabolic arcs.

26. In the preceding Example find the condition which must hold in order that after describing n parabolic arcs the particle should be again at the starting point.

27. A particle is projected with a given velocity at a given inclination to the horizon from a point in an inclined plane: find the whole time which elapses before the particle ceases to hop.

28. In the preceding Example find the condition which must hold in order that the particle may cease to hop just as it is again at the starting point.

29. In Example 25 find the cotangent of the inclination to the plane of the direction of motion of the particle at the beginning of the nth arc.

30. Shew that the time of descent to the lowest point of a very small circular arc is to the time of descent down its chord as the circumference of a circle is to four times its diameter.

P

5. 9 lbs. and 3 lbs.: 1 inch. 6. As 3 to 4.

7

8. of a cubic foot.

20

9. As 16 to 9.

II. 1. 64, 8. 2. 37. 3. 9, 12.
at an angle of 45° with the resultant.
9. A right angle. 10. 5, 5√√√3.

4. 3, 6. 6. 5/2lbs.,
8. As /3 is to 2.
11. In a straight line.

III. 2. By Art. 34, forces 1, 1, 1 are in equilibrium
and may be omitted; thus the resultant is equivalent to
that of forces 1 and 2 at an angle of 120°. 4. See Art. 37.
5. 15 lbs., 20 lbs. 6. 4 lbs. 7. Let OA and OB denote
the equal forces, OD their resultant; produce AO to C so
that ŎC=20A; and let OE be the resultant of OB and
OC: then it is given that OE=OD. The resultant of OE
and OD is equivalent to that of twice OB and half OC,
and is therefore equal to OE. 8. It follows from 7 that
the angle EOD = 120°; and the straight line which bisects
the angle EOD must make with OB an angle equal to
EOC the angle OED=the angle ODE. 12. The re-
sultant is 2/2 lbs., and it is parallel to a side of the
square. 13. The resultant coincides in direction with the
straight line from the point to the intersection of the
diagonals of the rectangle, and is equal to twice that straight
line. 14. Use the polygon of forces.
15. Use Ex. 14:
if n be the number of equal parts the resultant is repre-
sented by (n- times the radius.
R = nr

IV. 1. The resultant is 9/2 lbs.: it is parallel to the
diagonal AC; and it crosses AD at the distance AD

16

2

9

3. 13 inches from
9. Pa22, where

V. 7. Take moments round H: thus we find that KI is parallel to BC. 8. Take moments round an end of one force: thus we find that the other two forces are bisected at 0.

VI. 1. (20) lbs. 5. The angle ACB is given ; and since P, Q, and R are given, the angles which the direction of Ŕ makes with AC and CB are given. 6. See Art. 38, and Euclid, m. 21, 22. 8. The point must be at the intersection of the straight lines which join the middle points of opposite sides. 9. The forces 1 and 3 are at right angles; the forces 2 and 1 at 120o. 11. Let CD be the resultant of CA and CB. Let A come to a. Take Dd equal and parallel to Aa; then ad is equal and parallel to CB. Thus Cd is the resultant of Ca and CB.

5 inches from the end at which the force of 4 lbs. acts. 4. At a distance from the centre of the hexagon equal to one-fifth of a side. 5. At the point at which the force

of 8 lbs. acts.

6. At the distance

1 n- 1

of the radius

7. 6 inches from the end.

12. The

from the centre.
force at the point A must be Q+R−P; and so on.

IX. 2. One foot from the end. 3. Suppose the straight line parallel to BC; let D be the middle point of BC: the centre of gravity is on AD at the distanceAD from A. 4. At a distance from the centre of the larger circle equal to one-sixth of the radius. 5. Equal forces.