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Hence, 88 velos are added on in the course of each minute; that is,

4 velos are added on per second. Hence, i mile per min. per min. 89 celos. Therefore the point will move over

{22 x 60+1. 8x (60)2} feet in 60 secs.; that is,

(1320+ 2640) ft., or, 3960 ft.

EXAMPLES. VIII. The motion in each case is that of uniform acceleration in a straight line.

1. A point has initially 96 velos and it has – 32 celos; in how many seconds will it be at a distance 144 ft. from its initial position ?

2. A point has initially 20 velos, and it has 2 celos; how long will it take to go 100 ft.? and how far will it go in attaining a velocity of

25 velos?

3. One point P has initially 200 velos and it has 32 celos; a second point Q has initially 1000 velos and 16 celos; initially they start from the same point in the same direction; when are they again together? and how far will they have gone ?

4. A point starts from rest with 32 celos, and from the same place, at the same time and in the same direction another point starts with initially 40 velos and 30 celos; when and where will the first point overtake the second ? and when will they have equal velocities?

5. A point having initially 20 velos attains a velocity of 30 miles per hour in passing over 128 ft. ; find its acceleration.

6. A point after passing over a quarter of a mile with an acceleration of 20 ft. per second per second, attains a velocity of 2704 ft. per second ; what was its initial velocity?

7. A train in passing over a quarter of a mile increases its velocity from 5 miles an hour to 35 miles an hour ; what is its acceleration ?

8. A train moving at the rate of 45 miles an hour is stopped by the continuous brake in 220 yds. ; what is its 'acceleration ??

9. A train has a velocity of 60 miles per hour and it has a retardation of one mile per minute per minute ; how far will it go before it comes to rest ?

10. A point in passing over a mile has its velocity increased from a mile in 3 minutes to a mile in 23 minutes : how many celos has it?

11. A point passes over 100 yards in 10 seconds, and the next 100 yards in 12 seconds; when will it be 100 yards further on?

12. A point passes over 200 ft. in 12 seconds, and the next 300 ft. in 20 seconds; when will it be 400 ft. further on?

MISCELLANEOUS EXAMPLES. IX.

4

The motion in each case is uniform acceleration in a straight line.

1. A point moves over 7 ft., 9 ft., II ft., 13 ft. respectively in four consecutive seconds; find its average velocity; and supposing the velocity to be uniformly increasing, find its acceleration and its velocity at the beginning and at the end of the four seconds.

2. A point has 4 celos for 10 seconds and passes over 1000 ft. in those 10 seconds; find its initial velocity; and if it had celos from rest, find when and where it started from rest.

3. At the beginning of an interval of 12 seconds a point has 40 velos, and its average velocity for the 12 seconds is 80 velos; find its acceleration.

4. A point passes over 5 ft. in one second, 16 ft. in the next 2 secs., II ft. in the next second, 45 ft. in the next 3 seconds; shew that this is consistent with a constant acceleration, and find the acceleration.

5. A point has initially 20 velos and it has – 2 celos; find when it will be 100 ft. from the initial position.

6. A point has initially 20 velos and it has 32 celos; find how long it will take to go 1800 feet.

7. A point has initially 120 velos and it has – 32 celos; find when it will be 200 ft. from the starting point.

8. A point has 20 velos and it goes twice as far in the third and fourth seconds together as it does in the first two seconds of its motion; find the acceleration.

9. A point starts from rest with a constant acceleration ; shew that it goes in the second half of any interval three times as far as it does in the first half.

10. Two points move from the same point in the same line, one with a constant velocity 4 velos, the other from rest with 2 celos; they start together; when will they be together again?

11. A point P starts along a line with constant acceleration a celos, and t seconds later a point Q starts along the same line with constant velocity u velos; prove that will not overtake P unless u is greater than at.

12. The line AB is 192 ft. long; P starts from A along AB with 8 velos and moves uniformly; Q starts from rest at B along BA with 8 celos; find where they will meet.

13. Two points P and Q have each 32 celos in the same direction AB; P starts from A from rest and starts simultaneously from B having a velocity towards B just sufficient to carry it to B; find where they pass each other, if AB is 120 ft.

14. Two points P and Q have each g celos in the same direction AB; P starts from A from rest and starts simultaneously from B (which is a ft. from A) having a velocity of /(2ag) velos towards B; find where they pass each other.

15. Two points P and Q have each g celos in the same direction, say vertically downwards; P starts from rest from a point A and I starts from B, vertically below A, with a certain velocity v celos upwards. Shew that however small v may be, P will always overtake l.

16. A point moves from rest over a certain distance with 32 celos; it describes of the whole distance during the last second of the motion; find the whole interval occupied.

17. A point has initially mn velos and - n celos; shew that after (n+k) seconds it returns to the same point which it was passing at the end of the first (n - k) seconds, with the same number of velos.

18. Prove that the distances passed over in successive equal intervals by a point starting from rest and moving with a constant acceleration are proportional to the series 1, 3, 5, 7...

19. A point having a certain velocity, moves with an acceleration in the opposite direction, until it comes to rest; prove that the intervals of describing the first half and second half of its path respectively are as 1 : V2+1.

20. A point having a celos, passes over h feet in a certain interval, with an average velocity u velos, and its velocity is increased during the interval by v velos; prove that uv=ah.

21. An engine-driver, whose train is travelling at the rate of 30 miles an hour, sees a danger signal at the distance of 220 yards, and does his best to stop the train; supposing that he can stop the train when travelling 30 miles an hour in 440 yards, shew that his train will reach the danger signal with a velocity of 2n} miles per hour nearly.

22. A point, moving with a constant acceleration a celos, passed over twice as many feet in a certain interval t seconds as it did in the immediately preceding interval of t seconds; shew that its velocity at the beginning of the first interval was žta velos.

MASS AND FORCE.

27. Consider some definite quantity of matter; say, a cubical lump of iron. This (i) consists of a certain kind of material (iron), (ii) has a definite shape, (iii) has a definite volume, (iv) contains a certain quantity of matter, (v) has a definite weight.

Each of these (i), (ii), (iii), (iv), (v) is a separate and distinct idea applicable to the lump of iron.

In dynamics we are chiefly concerned with the last two; viz., the quantity of matter, which we shall call the mass, and the weight

We proceed to consider what is meant by mass.
28. We may illustrate the idea of mass as follows:

Let us imagine a perfectly smooth, perfectly horizontal sheet of icet. Upon this, place a lump of matter, such as a smooth block of stone in the form of a cubic foot. Now urge the lump forward with a horizontal pressure or force, taking care to apply the force constantly and uniformly for an interval of, say, 10 seconds. What happens ? It will be found that by the force the lump has a certain amount of acceleration given to it; so that a velocity grows in it as long as the force is applied ; and at the end of the 10 seconds it will have acquired a certain number of velos. At the end of the 10 seconds let the force be withdrawn. What happens? The lump has a certain number of velos; and it will be found that it will continue to move uniformly with that number of velos, so long as there is nothing in the nature of a force applied to it.

29. Suppose now that we try the experiment of Art. 28 with several lumps of the same size but of different material; say, one of lead, one of stone, one of cork; and suppose that we can apply an equal constant horizontal force to each continuously for an interval of, say, 10 seconds. What happens? It will be found that the lead, the stone, and the cork, have each a constant number of celos communi

+ It must be noticed that these conditions can only be approximately fulfilled; the best ice is not perfectly smooth and the movement of the air would interfere to a certain degree with such an experiment.

30. DEF.

cated by the force; but that the number of celos given to the cork is greater than the number given to the stone, and the number given to the stone greater than the number given to the lead. So that at the end of the 10 seconds they will each have acquired a certain number of velos, the cork more than the stone, the stone more than the leadt; also, that after the force has been withdrawn, each will continue to move with its own constant velocity, so long as there is nothing in the nature of a force applied to it.

We shall choose as our unit mass the mass of a certain piece of metal [called 1 lb. (avoirdupois).]

We shall call this unit mass a pound, or, 1 lb.

31. DEF. Force is that which when applied to mass produces, or tends to produce, in it acceleration in the direction of the force; so that the force varies as the acceleration produced in a given mass; and also varies as the mass in which a given acceleration is produced.

Consider (i) the experiment of Art. 27; here, if we double the force, we shall find that we produce in the same mass double the acceleration; consider (ii) the experiment of Art. 28; here, if the stone has 5 times the 'mass' of the cork, we must have 5 times the force acting on the stone to produce in it the same number of celos as in the cork.

32. DEF. We shall choose as our unit force that force which acting on a pound produces in it i celo.

We shall call this unit force a poundal.

33. The statement of Art. 31 is to be given the fullest possible interpretation; it asserts that

I. Force is that which produces acceleration in mass; therefore, whenever a mass has acceleration, it is under the action of some external force.

II. When no external force acts on a mass it has no acceleration; in other words, if at rest, the mass remains

+ It is instructive to imagine the three blocks of the same size and shape and painted the same colour; the application of an equal horizontal force to each, will at once reveal which is lead, which stone and which cork.

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