Ex. 3. A balloon is 400 feet from the ground, and ascending at the rate of 10 feet per second. What time would a sandbag take to fall to the ground from it?

The sandbag has a velocity of 10 feet upwards per second, and when let fall it is subject to an acceleration of

32 ft. downwards per sec. per sec. ;

Ex. 4. St. Rollox stalk is. 445 ft. high; at what rate must a bullet be shot vertically upwards to reach the top (disregarding the resistance of the air) in a second? What would be its speed when passing the top? How high would it rise?

1st. Let the velocity of projection be

2nd. The original velocity is 461 ft. up per sec.; gravity acts at 32 ft. down per sec. = = sec.,

for 1 sec., 32 ft. down per sec.;

hence the velocity when passing the top is

Ex. 5. The speed of a railway train increases uniformly for the first three minutes after starting, and during this time it travels one mile. What speed, in miles per hour, has it now gained, and what space did it describe in the first two minutes ?

1. Express an acceleration of 500 centimetres per second per second in terms of the kilometre and minute.

2. The velocity and the acceleration of a moving point at a certain moment are both measured by 10 the foot and the second being the units of space and time. Find the numbers measuring them when the yard and the minute are the units. 3. On what arbitrary units does the numerical quantity g depend? If each unit becomes m times its former amount, what will be the new value of g?

4. Express 32.2 feet per second per second in terms of yard per minute per minute.

5. To ascertain the height of a precipice a stone was dropped from the edge and was observed to take three and a half seconds to reach the bottom. What is the height of the precipice?

6. A stone dropped from the top of a cliff is observed to reach the bottom in 6 seconds. Find the height.

7. A boy throws a stone vertically into the air with a velocity of 80 feet per second. How much time has he to escape from it returning?

8. A stone is let fall, and another is at the same instant projected upwards from a point 500 feet lower in the same vertical. With what speed must it be projected so that the two may meet half way?

9. How far has a body fallen from rest when it has acquired a velocity of (1) 20 feet per second, (2) 100 feet per second?

10. A stone, dropped from rest, falls under the action of gravity 65 feet during a particular second of time. How long before the end of this second did it begin to fall?

11. Two particles are let fall, the one from 100 feet, the other from 225 feet high, and they reach the ground at the same time. Find the interval between their times of starting.

12. A particle is dropped from a height; supposing it reaches the ground in 12 seconds, how much did it fall during the last second, and how far has it fallen altogether?

13. A stone is thrown downwards, and its average velocity for the second second of its fall is 21⁄2 times that for the first second. What was the initial velocity?

14. A rifle bullet is shot vertically downwards from a balloon at rest at the rate of 400 feet per second. How many feet will it pass through in two seconds, and what will be its velocity at the end of that time, neglecting the resistance of the air and estimating the acceleration due to gravity at 32?

15. A stone thrown vertically upwards strikes the ground after an interval of 10 seconds. With what velocity was it projected, and to what height did it rise? 16. If a body is projected upwards with a velocity of 120 feet per second, what is the greatest height to which it will rise, and when will it be moving with a velocity of 40 feet per second?

17. With what velocity must an arrow be shot vertically upwards in order that it may just reach the top of a stalk 150 feet high?

18. My watch beats five times each second. A boy throws a stone into the air vertically upwards, and I reckon 271⁄2 beats of my watch from the instant the stone leaves the boy's hand until it strikes the ground. Taking g = 32, show that the boy's hand when the stone left it was moving with a velocity of 88 feet per second; and find how high the stone went.

19. What is meant when it is said that the acceleration of the speed of a particle is 10, the units being foot and second? If the particle were moving at any instant at the rate of 71⁄2 feet per second, after what time would its speed be quadrupled? and what distance would it describe in that time?

20. A body describes distances of 120 yards, 228 yards, 336 yards, in successive tenths of a minute. Show that this is consistent with constant acceleration of its velocity, and find the numerical value of the acceleration if the units of time and distance are a minute and a yard.

21. A body is moving in a straight line with a uniform velocity of 10 feet per second. Suddenly a force begins to act upon it in a direction contrary to that of its motion, whose acceleration is 5 feet per second per second. In what sense and with what velocity will the body be moving at the end of 2 seconds from the moment the accelerating force began to act?

22. A particle is found to be moving in a straight line at the rate of 5 feet per second, a quarter of a minute afterwards at the rate of 50 feet per second, half a minute afterwards at 95 feet per second. Show that this is consistent with a constant rate of change of speed, and find its value.

SECTION XXI.—ACCELERATION.

Acceleration being a

ART. 120.-Rate of Change of Velocity. vector quantity, is resolved and compounded in the same manner as a simple vector or as a velocity. When our attention is restricted to one plane, and to rectangular components, the equivalence between the acceleration and its components is expressed by aL along per T per Ta1L adj. per T per T + a.L opp. per T per T. When the components are rectangular, the numbers a, a, a,, are connected by the condition

a2 = a2 + a.

From this equivalence partial equivalences may be derived, as in Art. 111.

The cosine of the direction of the acceleration is given by

change of velocity can be resolved into rectangular components is by taking the direction of motion for the time being as one line, and the perpendicular to it as the other line. The former com

ponent is the rate of change of speed, which has been already considered; there remains for consideration the component along the transverse line. It does not affect the speed, but it alters the direction of the motion. It is proportional to the square of the velocity of the point and to the curvature produced. The dependence is fully expressed by

1 L per T per T = (L per T)2 by (radian per L arc). By transforming the right hand unit we obtain equivalent forms, 1 L per T per T= (L per T)2 per L radius (Art. 74),

= (L arc per L radius per T) by (L per T),

= (radian per T) by (L per T).

ART. 122.-Dimensions.

The dimensions of the several units
The dimension of a

expressed above are said to be the same.
unit with respect to a fundamental unit as L is reckoned by taking
the number of times it enters directly and the number of times it
enters inversely and taking the difference. In transforming from
one set of fundamental units to another it is this difference upon
which the transformation depends. The dimensions of V are 3
with respect to L; of L per T, 1 with respect to L, and 1 with
of L per T per T, 1 with respect to L, and
respect to T;
2 with
respect to T.

ART. 123. Simple Harmonic Motion. Let a point move round a circle with uniform angular velocity, then the component of this motion along any diameter of the circle is a simple harmonic motion.

Let the point Q move round the circle O, whose radius is a L, with a uniform angular velocity w radian per T. Let AA' be the line of the simple harmonic motion, and suppose that t have elapsed since Q was at A.

First, the position of P. The point Q will then be at an angle of of radians, and the component along AA' of the vector to Q will be a cos(wt) L (Fig. 15).