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3. If we wish to give any one a clear idea of the magnitude of some physical quantity, we describe it as bearing such and such a ratio to some definite arbitrarily chosen amount of that quantity, known to him. The known definite amount is called the unit of the physical quantity generally, while the ratio is called the numerical measure, or simply the measure of the particular amount under consideration.

If for instance, the area of a certain field be 12 acres, and an acre be chosen as the unit area, the ratio of the area of the field to that of the unit is 123, which is therefore the numerical measure of the area of the field.

We shall suppose then, that we have fixed on some particular length as the unit length, and some particular interval of time as the unit of time.

If the velocity of a point be uniform, its numerical measure is the numerical measure of the distance traversed by it during the unit of time. It may happen that the point's velocity, though uniform for a finite interval of time, is not so for the unit of time: in that case, its numerical measure is that of the space the point would traverse during the unit of time, provided it moved throughout with the same velocity as during the finite time. The velocity which we call the unit velocity, or whose numerical measure is one, is the velocity of a point which traverses the unit of length in the unit of time.

Def. The mean or average velocity of a point during any interval of time is the velocity with which a point, moving uniformly during that time, would describe the same distance. Its numerical measure is therefore the numerical measure of the distance described, divided by that of the time required.

Def. The velocity of a point at any instant, is the limit of the mean velocity of the point during an interval of time including the particular instant, when the interval is diminished indefinitely.

Ex. 1. Compare the velocities of two points which move uniformly, one through 5 feet in half a second, and the other through 100 yards in a minute. Ans. 2: 1. Ex. 2. A railway train travels 160 miles in 6 hours 30 minutes. What is its average velocity in feet per second? Ans. 36.1 nearly,

Ex. 3. One point moves uniformly twice round the circumference of a circle, while another moves uniformly along the diameter: compare their velocities. Ans. 2π: 1. Ex. 4. A fly-wheel is 14 feet in diameter, and is observed to go round uniformly fifteen times in a minute: find the velocity of a point in the circumference. Ans. 11 feet per second nearly.

Ex. 5. Supposing the earth to rotate about its axis in 23 hours 56 minutes, its equatorial diameter being 7925 miles, find the velocity of a point at the equator relative to the earth's centre, in feet per second. Ans. 1526 nearly.

4. Now a velocity is entirely known, if its direction and magnitude are known. But as a straight line AB can be drawn in any direction, it can be drawn so as to indicate fully the direction of a point's velocity, provided we shew either by an arrow-head or by the A order of the letters AB, the sense of the velocity, i.e. whether its

Fig.!

direction be from A to B or from B to A. As we can make the line of any length, we can make it so that its length bears the same ratio to some arbitrarily chosen length as the velocity considered bears to the unit of velocity. If this be done, and we know the scale, i.e. the length chosen to represent the unit velocity, the line AB will also represent the magnitude of the velocity considered.

5. A point may be moving with several independent velocities at once for instance, we know that the earth as a whole is describing an orbit about the sun, and that all points on the earth's surface are describing circles about the earth's axis; if then, a point be moving on the earth's surface, it has relatively to the sun, three independent velocities, viz. its velocity on the earth's surface, the velocity of the point

of the earth's surface it occupies at the particular instant, relatively to the centre of the earth, and the velocity of the earth's centre about the sun.

Def. When a point has several independent velocities, the single velocity which would alone give the point's motion is called the resultant of the other velocities.

Let us consider the case of a point moving in a straight line along the deck of a ship, with uniform velocity relative to the ship, which is sailing with uniform velocity in a straight line along the earth's surface. It is required to find the point's motion relative to the earth's surface, i.e. given its position at one instant, it is required to find its position at the end of a given time. Now since the point's motion on the ship's deck is entirely independent of the ship's motion, if we suppose the point fixed to the deck during the time considered, so that its motion is that of the ship, then the ship to remain stationary while the point moves for an equal time along the deck with its velocity relative to the ship, the final position of the point will be the same as if the two motions had taken place simultaneously, as they really do.

The above illustration exemplifies a general axiomatic principle, which may be stated thus: if during a certain time a point has several independent motions, its actual position at the end of any portion of that time may be found by imagining that all the motions take place separately during a number of successive periods of time equal to the one considered, instead of supposing that all the motions take place simultaneously, which is what really takes place. Of course the imaginary motion only gives the same initial and final positions of the point as the real one, and not in general intermediate ones, although by taking the periods of time very small, but very large in number, the imaginary motion which gives us the real position of the point at the end of each of them, will give us an infinite number of points on the point's actual path. The motions referred to above are not of necessity due to uniform velocities.

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The Parallelogram of Velocities.

6. If the two independent velocities of a point be represented in magnitude, direction and sense by two straight lines drawn from, (or to) a point, and a parallelogram be constructed on them as adjacent sides, the resultant velocity is represented in magnitude, direction and sense by the diagonal drawn from (or to) the point of intersection of these sides.

Let the lines OA, OB represent in magnitude, sense and direction the velocities u, v of the point: complete the parallelogram OACB, and join OC; then OC shall represent the resultant velocity. If O be taken as the

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initial position of the point, its position at the end of a time t can be found by supposing that it first moves with the velocity u for a time t, and then with the velocity v for the same time. If it moves with the velocity u alone, it will at the end of a time t be at a in the line OA, where Oaut; if now it moves with the velocity v alone for a time t, it will arrive at c, where ac is parallel to OB, and ac vt. c then is the position of the point at the end of a time t, when the motions take place simultaneously.

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But because ac: Oa AC: OA, c is in OC, i.e. OC represents the direction of the resultant velocity. Also the magnitude of the resultant velocity is to the velocity OA as Oc is to Oa, i.e. as OC: OA: hence OC represents the magnitude of the resultant velocity. The sense of the resultant velocity is clearly OC.

The above proposition holds at any instant, even though the independent velocities be varying velocities: for it is only necessary to suppose that the time t is ultimately indefinitely small, and the above proof holds.

Ex. 1. If a boat can steam 9 miles an hour up stream, and 13 miles an hour down stream: find the velocity of the stream.

Ans. 2 miles per hour. Ex. 2. Velocities of 4 feet and 16 feet per second in directions at right angles to each other are simultaneously communicated to a body: determine the resultant velocity. Ans. 16.49 feet per second.

Ex. 3. A ship whose head points N.E. is steaming at the rate of 12 knots an hour in a current which flows S.E. at the rate of 5 knots an hour, find the velocity of the ship relative to the sea bottom.

Ans. 13 knots an hour.

7. All the objects around us that we can see and touch, and even invisible substances, such as air, are material bodies or composed of matter. The various properties of matter, such as hardness, density, &c., can be investigated, but no definition of matter can be given which would give any idea of it to a being that had had no experience of it.

Any limited portion of matter is called a Material Body or simply a Body. When we consider a body whose dimensions are so small that we are only concerned with its motion as a whole, and not with any rotational motion it may have, we describe it as a material particle, or simply a particle.

The term Mass is synonymous with the phrase Quantity of Matter, so that the mass of a body means the quantity of matter in it.

If two bodies are composed of the same substance under the same conditions we can compare their masses by comparing their volumes. Thus a quart of water contains twice as much matter as a pint. Liquids generally are sold by volume. In the case of solids it is often difficult to determine their volumes and when

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