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Let AB, AC represent the coexisting velocities. At the end of an unit of time the point will have moved over the space AB, but the line AB has then arrived at the position CD, therefore the point will be found at D, the opposite angle of the parallelogram BC.

Let ab be the position of the line AB, and d the position of the moving point in it, at any other instant of the motion. Join AD, Ad.

Then ad CD :: time required for the point to move over ad : time required for it to move over CD

(because the motion is uniform),

i.e. as the time in which the extremity of the line ab moves over Aa: the time in which it moves over AC,

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therefore the triangles Aad, ACD have equal angles at a, C, and the sides containing them proportional; therefore they are equiangular, i. e. the angles aAd, CAD are equal;

therefore d is in the diagonal AD, and consequently the moving point passes along AD.

Also it describes AD uniformly,

for Ad: AD: Aa: AC

:: time in which the extremity of ab passes over Aa

: time in which it passes over AC

:: time in which the moving point passes in space over Ad

: time in which it passes over AD;

.. AD represents the resulting velocity in magnitude and direction (Art. 7).

9. This may be extended to space of three dimensions, as follows:

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If OA, OB, OC represent in magnitude and direction three coexistent velocities, OD the diagonal of the parallelepiped constructed on OA, OB, OC as edges, will represent the resulting velocity both in magnitude and direction.

For the velocities OA and OB are equivalent to a velocity OE, and therefore ОA, OB, OC are equivalent to OE and OC, and consequently to OD.

This proposition is called the parallelepiped of velocities.

10. From this it is clear that we may consider any given velocity to result from the coexistence of other velocities in two or three given directions, according as we deal with space of two or three dimensions, and this is called resolving a velocity.

It is usual to resolve into directions at right angles with each other, and in that case if OC represent the velocity v, and the angle CO = 0,

the velocity in Ox will be represented by OA, i. e. OC cos 0, and therefore = v cos 0;

and the velocity in Oy will be represented by OB, and therefore = v sin 0.

y

C

B

A

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If we deal with space of three dimensions, OD representing the velocity v, and the angles DOx, DOy, DOz, being called a, B, Y,

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These are called the resolved parts of the velocity v.

11. Conversely, having given the velocities in given directions in space, we can find an analytical expression for the resulting velocity in magnitude and direction. This is called compounding the given velocities.

For if in space of two dimensions, the velocities along Ox and Oy be given v1, v2, respectively, we have

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y

C

B

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Or otherwise, v1 = v cos 0, v2 = v sin 0; whence the same results are obtained.

If we consider space of three dimensions, and the velocities along Ox, Oy, Oz be given v1, V2, V37

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B

Or otherwise, v cos a = v1, v cos ẞ= v2, v cos y = v1, whence, as by a well-known theorem, cos'a + cos2ß + cos3y = 1, the same results are obtained.

12. If v be the velocity of a point in the direction of a line whose direction-cosines are l, m, n, the resolved part of this in the direction of another line whose direction-cosines are l', m', n', will equal

vx cosine of the angle between the lines

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lv, mv, nv, are the resolved parts of the given velocity in the directions of the co-ordinate axes, and as l', m', n', are the cosines of the angles between these and the second line, this proposition shews that the resolved part required may be correctly obtained

by resolving the given velocity into three directions at right angles to each other, and taking the sum of these resolved separately upon the second line.

13. We have hitherto considered velocity as uniform: but it is plain that the velocity of a moving point may be continually changing, and we must fix a measure for such a velocity.

If another point be moving uniformly with that velocity with which the proposed point is moving at the instant under consideration, this velocity is measured by the space which the point passes over in an unit of time, and consequently we must measure the varying velocity by the space which would be passed over in an unit of time, supposing that the velocity were to remain constant for that unit of time.

14. We shall now obtain a relation between the space, time, and velocity in such a case.

If s be the space passed over in time t reckoned from the instant under consideration, and if s' be the space which would be described in time t supposing the velocity constant, we have as before (Art. 4),

v =

and this is true always, whatever be the magnitudes of s' and t; therefore it is true when s' and t are indefinitely diminished, in which case

v = Limit === Limit = ;

t

because s and s' are ultimately equal.

A. 15. If we employ the differential calculus, we shall obtain the same result. For if a space ds be described in a small time St immediately subsequent to the instant under consideration, and if v', v" be the greatest and least values of the velocity during the time St, it is plain that ds < v'dt, and > v'dt;

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