Ex. 3.

smooth fixed pulley and P descend; after P has described a given space a, let a weight p be removed from P, leaving the remainder P-p less than Q. Determine the subsequent motion.

34. A weight P after falling freely through a feet, begins to raise up a weight Q>P, connected together by means of a string passing over a smooth fixed pulley; find the extreme height to which Q can rise, and the time of its ascending.

35. Find the straight line of quickest descent from a given point within a circle to the circumference.

36. Find the straight line of quickest descent, from the focus to the curve of a parabola, whose axis is vertical.

37. The major axis of an ellipse is vertical; determine the radius vector measured from the upper focus, down which the time of descent is the least possible.

MOTION UPON A CURVE, AND THE SIMPLE PENDULUM.

If l=number of inches in the length of a simple pendulum,

t sec. time of one oscillation;

If, in the latitude of London, L be the length of the seconds' pendulum, and g the force of gravity;

L=39'1393 in.;

log L= 1592613;

Ex. 4.

g=3219084 ft.; log g= 1507732;

T=3'14159 log = 497150.

1. Three planes A, B, C are in contact; A is vertical, B and C are inclined to the horizon at angles 60° and 30° respectively; find the velocity, with which a body beginning to descend from A will begin to move along the horizontal plane passing through the lower extremity of the plane C.

2. A ball having descended to the lowest point of a circle through an arc whose chord is a, drives an equal ball up an arc whose chord is b; find the common elasticity e of the two balls.

3. If be the angular distance of a body from the lowest point in a circular arc; show that the force in the direction of the arc is to the force in the direction of the chord as 2 cos 0: 1.

4. Having given the length of the seconds' pendulum, find the length of a pendulum that will oscillate 4 times in one second; and another 9 times in one minute.

5. In what time would a pendulum, 80 inches long, vibrate at the distance of two of the earth's radii, above the surface of the earth?

Ex. 4.

6. Find the length of a pendulum which oscillates as often in one minute as there are inches in its length.

7. Find the length of a pendulum that would oscillate three times, whilst a heavy particle falls from rest through 81 feet.

8. If a pendulum vibrate seconds at the earth, it would vibrate minutes at the moon, the distance of the moon from the earth's centre being taken equal to 30 times the earth's diameter.

9. A seconds' pendulum is lengthened 105 inches; find the number of seconds it will lose in 12 hours.

10. A pendulum which should beat seconds, is found to lose 20 seconds a day. Determine the quantity by which its length should be increased or diminished.

11. A pendulum gains 3 seconds in an hour, before it is carried up a high mountain; at what height in the ascent would the pendulum keep true time, if the earth's radius were 4000 miles?

12. How high must a seconds' pendulum be carried above the level of the sea, that it may vibrate 598 times in 10 minutes, the radius of the earth being 3958 miles?

13. A seconds' pendulum is carried to the top of a mountain and there loses 48.6 seconds in a day; determine the height of the mountain, supposing the earth's radius to be 4000 miles.

14. The length of a pendulum that vibrates sidereal seconds being 38.926 inches; find the length of a sidereal day. Find the increment of the length of the pendulum, that it may measure mean solar time. 15. A pendulum, which would oscillate seconds at the equator, would, if carried to the pole, gain 5 minutes a day; compare the forces of equatoreal and polar gravity.

16. Two pendulums, the lengths of which are L and 7, begin to oscillate together, and are again coincident after n oscillations of L. Given L the greater, to find l.

17. Two pendulums, A and B, begin to oscillate together, and are again coincident after 12 oscillations of A; find the length of B, that of A being 38.3 inches and longer than B.

18. A pendulum, which vibrates seconds at Greenwich, taken to another place is found to lose n seconds a day; compare the forces of gravity at the two places.

19. If a clock, at a place A on the earth's surface, keeps true time, and when taken to another place B loses n minutes daily, but goes right on being shortened by the mth part of an inch; find the length of the pendulum.

20. A pendulum 40 inches long oscillates 3.5 times between the time of seeing the flash and hearing the report of a cannon; find the distance of the cannon from the observer, the velocity of sound being assumed equal to 35g per second.

21. If from the extremity of the vertical diameter of a circle the

Ex. 4.

chord of 60° be drawn ; compare the time of falling down this chord with the time of oscillation of a pendulum equal in length to the chord.

In the interior of the earth

Gravity varies directly as the distance from the earth's centre.

22. A seconds' pendulum, on being carried to the bottom of a mine, is found to lose 10 seconds a day; determine the depth of the mine, if the earth's radius be 4000 miles.

23. How far below the earth's surface, or how high above it, must the pendulum, whose length is 39 12 inches, be taken to oscillate seconds, the earth's radius being 3958 miles ?

24. If a pendulum, when carried to the top of a mountain, is observed to lose in a given time just twice as much as it does when taken to the bottom of a mine in the neighbourhood; show that the height of the one is equal to the depth of the other.

25. The times of oscillation of a pendulum are observed at the earth's surface, and at a given depth below the surface; hence determine the radius of the earth, supposed spherical.

26. A particle, acted on by gravity, descends from any point in the arc of an inverted cycloid, of which the axis is vertical, to the lowest point of the curve; find the time of descent.

PROJECTILES IN A NON-RESISTING MEDIUM.

If two straight lines be drawn through the point of projection, one horizontal, the other vertical; and these lines be taken for the coordinate axes of x and y respectively; V the velocity of projection, and a the angle which the direction of projection makes with the axis of x ; h the space due to the velocity of projection; then the equation to the curve described by the projectile is

If R be the horizontal range, T the time of describing this range, H the greatest height, then

2. T

2V
9

sin a. 3. R=2h sin za.

4. H=h sin2 a.

If R', T' be the range and time respectively on an inclined plane passing through the point of projection, and having an elevation ; then

If w be the weight of a ball or shell, p the weight of the gunpowder

used in firing the ball or shell from a mortar, and v the velocity generated by the powder, then

1. A body is projected in a direction making an angle of 15° with the horizon, with a velocity of 60 feet per second; find its range, greatest altitude, and time of flight.

2. A body is projected at an angle of 45°, and descends to the horizon at a distance of 500 feet from the point of projection; with what velocity was it projected, what was its greatest altitude, and the whole time of flight?

3. The horizontal range of a projectile is 1000 feet and the time of flight is 15 seconds; find the direction and velocity of projection; also the greatest altitude of the body during the flight.

4. The horizontal range of a body, projected at an angle of 150, is 841 feet; find how high the body would rise, if projected vertically upwards with the same velocity of projection.

5. If the horizontal range of a projectile be to the greatest height as 4: 3; find the angle of projection.

6. If the horizontal range of a body, projected with a given velocity, be three times the greatest altitude, find the angle of projection. Find this angle when the range is equal to the altitude. 7. Find the velocity and direction of projection of a ball, that may be 100 feet above the ground at the distance of half a mile and may strike the ground at the distance of 1200 yards.

it

8. From one extremity of the base (500 feet) of an isosceles triangle, whose vertical angle is 36°, situate in a vertical plane, a body is projected in the direction of the side adjacent to that extremity, so as to strike a body placed in the other extremity; find the velocity of projection, and the time of flight.

9. A shell being discharged at an angle of 45°, its explosion was heard at the mortar 3 seconds after the discharge; required the horizontal range, the velocity of sound being 35g per second.

10. A ball is fired at a given elevation a towards a person who is on the same horizontal plane as the gun; if the ball and the sound of the discharge reach him at the same instant, find the range; the velocity of sound being 359 per second.

11. If a body be projected with a velocity of 850 feet per second in a direction making an angle of 60° with the horizon; find the focus of the parabola described, and also its latus rectum.

12. Find the velocity and direction, with which a body must be projected from a given point, that it may hit two other given points in the same vertical plane.

13. If the area of the parabola described by a projectile be one

Ex. 5.

third of the square described on the horizontal range; find the angle of projection.

14. Two bodies are projected from the same point with the same velocity; the directions of projection are measured by the angles a and 22 respectively; compare the areas of the parabolas described, supposing the horizontal ranges equal.

15. If the areas in the last problem be equal; what is the value of a? 16. Two bodies are projected from the same point with equal velocities, and their horizontal ranges are equal; if the areas of the parabolas described are as 2 : 1; find the angles of projection.

17. A shot is fired with a given velocity towards a tower whose horizontal distance from the cannon is one-half the range, and whose altitude subtends an angle tan-13 at the point of projection; find the inclination of the cannon to the horizon, that the shot may strike the summit of the tower.

18. At a distance a from the bottom of a vertical line, a ball is projected at an angle of 45°, which just touches the top, and afterwards strikes the ground at the distance b from the bottom on the other side; find the height of the line.

19. A body is projected at an angle of 60° elevation, with a velocity of 150 feet per second; find the direction and velocity of the projectile after the lapse of 5 seconds; also its height above the horizontal plane passing through the point of projection.

20. Two bodies are projected from the same point with equal velocities so as to describe the same horizontal range, and the times of flight are as 3: 1; required the directions of projection.

I

21. If two bodies be projected from the same point with equal velocities, and in such directions that they both strike the same point on a plane whose inclination to the horizon is ß; if a be the angle of projection of the first, compare the times of flight.

22. AB is the vertical diameter of a circle. A perfectly elastic ball descends down the chord AC, and being reflected by the plane BC, describes its path as a projectile; show that the body will strike the circle at the opposite extremity of the diameter CD.

23. A ball of given elasticity is projected with a given velocity at a given elevation. On meeting the horizontal plane it rebounds, describes another parabola, and again rebounds; and so describes a series of parabola. Find the whole horizontal distance described before the ball ceases to rebound.

24. The velocities at the extremities of any chord of the parabola described by a projectile, when resolved in a direction perpendicular to the chord, are equal.

25. A body projected from the top of a tower, at an elevation of 30° above the horizon, fell in n seconds at a distance of a feet from the base; find the height of the tower.