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mean : and hence its value at the middle of the interval must be the mean of its first and last values. In symbols; if at time t=0 the velocity was V, then at time i'it is
v= V + at. Also the space (s) described is equal to the product of the time by the average velocity. But we have just shown that the average velocity is
=$(V+V + at) - V + fat, and therefore
* Vt. + fat. Hence, by algebra,
V + 2ax = 1 + 2 Vat+ dt =(V+af)* - , Or
1 2 - 1V) = axa If there be no initial velocity our equations become
0.= at, *= fat', 17* = ax.; V2=2aX Of course the preceding formulae apply to a constant retardation, as
in the case of a projectile moving vertically upwards, by simply giving a a negative sign.
44. When there is uniform acceleration in a constant direction, the path described is a parabola, whose axis is parallel to that direction. This is the case of a projectile moving in vacuo.
For the velocity (V) in the original direction of motion remains unchanged; and therefore, in tiine t, a space Vt is described parallet to this line. But in the same interval, by the above reasoning, we see that a space fat is described parallel to the direction of acceleration Hence, if AP be the direction of motion at A, AB the direction of acceleration, and the position of the point at time, t;
draw QP parallel to BA, meeting AP in
AP= , PQ-fat.
and is therefore known. Also QA is known in direction, for 4P bisects the angle, OAC, between the focal distance of a point and the diameter through it.
45. When the acceleration, whatever (and however varying) be its magnitude, is directed to a fixed point, the path is in a plane
passing through that point; and in this plane the areas traced out by the radius-vector are proportional to the times employed.
Evidently there is no acceleration perpendicular to the plane containing the fixed point and the line of motion of the moving point at any instant; and there being no velocity perpendicular to this plane at starting, there is therefore none throughout the motion ; thus the point moves in the plane. For the proof of the second part of the proposition we must make a slight digression.,
- 46. The Moment of a velocity or of a force about any point is the product of its magnitude into the perpendicular from the point upon its direction. The moment of the resultant velocity of a par. ticle about any point in the plane of the components is equal to the algebraic sum of the moments of the components, the proper sign of each moment depending on the direction of motion about the point. The same is true of moments of forces and of moments of momentum, as defined in Chapter II.
First, consider two component motions, AB and AC, and let AD be their resultant ($ 31). Their half-moments round the point o are respectively the areas OAB, OCA. Now OCA, together with half the area of the parallelogram CABD, is equal to OBD. Hence the sum of the two half-moments together with half the area of the parallelogram is equal to AOB together with BOD, that is to say, to the area of the whole figure OABD. But ABD, a part of this figure, is equal to half the area of the parallelogram; and therefore the remainder, OAD, is equal to the sum of the two half-moments. But OAD is half the moment of the A
B resultant velocity round the point O. Hence the moment of the resultant is equal to the sum of the moments of the two components. By attending to the signs of the moments, we see that the proposi. tion holds when 0 is within the angle CAB.
If there'be any number of component rectilineal motions, we may compound them in order, any two taken together first, then a third, and so on; and it follows that the sum of their moments is equal to the moment of their resultant. It follows, of course, that the sum of the moments of any number of component velocities, all in one plane, into which the velocity of any point may be resolved, is equal to the moment of their resultant, round any point in their plane. It follows also, that if velocities, in different directions all in one plane, be successively given to a moving point, so that at any time its velocity is their resultant, the moment of its velocity at any time is the sum of the moments of all the velocities which have been successively given to it.
47. Thus if one of the components always passes through the point, its moment vanishes. This is the case of a motion in which the acceleration is directed to a fixed point, and we thus prove the second theorem of $ 45, that in the case supposed the areas described,
by the radius-vector are proportional to the timess for, as we have seen, the moment of the velocity is double the area traced out by the radius-vector in unit of time.
48. Hence in this case the velocity at any point iş inversely as the perpendicular from the fixed point to the tangent to the path or the momentary direction of motion.
For the product of this perpendicular and the velocity at any instant gives double the area described in one second about the 'fixed point, which has just been shown to be a constant quantity.
As the kinematical propositions with which we are dealing have important bearings on Physical Astronomy, we enunciate here Kepler's Laws of Planetary Motion. They were deduced originally from observation alone, but Newton explained them on physical principles and showed that they are applicable to comets as well as to planets.
I. Each planet describes an Ellipse (with comets, this may be any Conic Section) of which the Sun occupies one focus.
II. The radius-vector of each planet describes equal areas in equal times.
III. The square of the periodic time [in an elliptic orbit] is pro portional to the cube of the major axis.
Sections 45–47, taken in connexion with the second of these laws, show that the acceleration of the motion of a planet:or comet is along the radius-vector.
49. If, as in $ 35, from any fixed point, lines be drawn at every instant representing in magnitude and direction the velocity of a point describing any path in any manner, the extremities of these lines form a curve which is called the Hodograph. The fixed point from which these lines are drawn is called the hodographic origin. The invention of this construction is due to Sir W. R. Hamilton; and one of the most beautiful of the many remarkable theorems to which it leads is this : The Hodograph for the motion of a planet or comet is. always a circle, whatever be the form and dimensions of the orbit. The proof will be given immediately.
It was shown (8 35) that an arc of the hodograph represents the change of velocity of the moving point during the corresponding time; and also that the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph is equal to the amount of the acceleration of the moving point.
When the hodograph and its origin, and the velocity along it, or the time corresponding to each point of it, are given, the orbit may easily be shown to be determinate.
(An important improvement in nautical charts has been suggested by Archibald Smith'. It consists in drawing a curve, which may be called the tidal hodograph with reference to any point of a chart for which the tidal currents are to be specified throughout the chief tidal period (twelve lunar hours). Numbers from I. to XII. are placed at marked points along the curve, corresponding to the lunar hours.
: Proc. R. S. 1865.
Smith's curve is precisely the Hamiltonian hodograph for an imaginary particle moving at each instant with the same velocity and the same direction as the particle of fluid passing, at the same instant through the point referred to.)
50. In the case of a projectile ($ 44), the horizontal component of the velocity is unchanged, and the vertical component increases uniformly. Hence the hodograph is a vertical straight line, whose distance from the origin is the horizontal velocity, and which is described uniformly.
51. To prove Hamilton's proposition ($ 49), let APB be a portion of a copic section and s one focus. Let P move so that SP describes equal areas in equal times, that is ($ 48), let the velocity be inversely as the perpendicular
-B SY from S to the tangent to the orbit. If ABP ТР be an ellipse or hyperbola, the intersection of the perpendicular with the tangent lies in the circle
U is, SZ is proportional to the velocity at P. Also SZ is perpendicular to the direction of motion PY, and thus the circular locus of Z is the hodograph turned through a right angle about S in the plane of the orbit.. If APB be a parabola, AY is a straight line. But if another point U be taken in YS produced, so that YS.SU is constant, the locus of U is easily seen to be a circle. Hence the proposition is generally true for all conic sections. The hodograph surrounds its origin (as the figure shows] if the orbit be an ellipse, passes through it when the orbit is a parabola, and the origin is without the hodograph if the orbit is a hyperbola.
52. A reversal of the demonstration of $ 51 shows that, if the acceleration be towards a fixed point, and if the hodograph, be a circle, the orbit must be a conic section of which the fixed point is a focus.
But we may also prove this important proposition as follows: Let A be the centre of the circle, and the hodographic origin. Join OA and draw the perpendiculars
PM to OA and ON to PA. Then OP is the velocity in the orbit : and ON, being parallel to the tangent at P, is the direction of acceleration in the orbit; and is therefore parallel to the radius-vector to the fixed point about which there is equable) description of areas. The velocity parallel to the radius-vector is therefore ON, and
Ñ the velocity perpendicular to the fixed line
ON OA OA is PM. But
Hence, in the orbit, the velocity along the radius-vector is proportional to that perpendicular to a fixed line: and therefore the radius-vector of any point is proportional to the distance of that point from a fixed line-a property belonging exclusively to the conic sections referred to their focus and directrix.
53. The path which, in consequence of Aberration, a fixed star seems to describe, is the hodograph of the earth's orbit, and is therefore a circle whose plane is parallel to the plane of the ecliptic.
54. When a point moves in any manner, the line joining it with a fixed point generally changes its direction. If, for simplicity, we consider the motion to be confined to a plane passing through the fixed point, the angle which the joining line makes with a fixed line in the plane is continually altering, and its· rate of alteration at any Sinstant is called the Angular Velocity of the first point about the second. If uniform, it is of course measured by the angle described in unit of time; if variable, by the angle which would have been described in unit of time if the angular velocity at the instant in. question were maintained constant for so long. In this respect the process is precisely similar to that which we have already explained for the measurement of velocity and acceleration.
We may also speak of the angular velocity of a moving plane with respect to a fixed one, as the rate of increase of the angle contained by them; but unless their line of intersection remain fixed, or at all events parallel to itself, a somewhat more laboured statement is required to give a complete specification of the motion.
55. · The unit angular velocity is that of a point which describes, or would describe, unit angle about a fixed point in unit of time. The usual unit angle is (as explained in treatises on plane trigonometry) that which subtends at the centre of a circle an arc whose length is equal
180° to the radius; being an angle of = 57°29578 ... = 57°17'44":8 nearly
56. The angular velocity of a point in a plane is evidently to be found by dividing the velocity perpendicular to the radius-vector by the length of the radius-vector.
57. When the angular velocity is variable its rate of increase, or diminution is called the Angular Acceleration, and is measured with reference to the same unit angle.
58. When one point describes uniformly a circle about another, the time of describing a complete circumference being T, we have the anglé 20 described uniformly in, 7'; and, therefore, the angular velocity is F. Even when the angular velocity is not uniform, as in 'a planet's motion, it is useful to introduce the quantity **, which is then called the mean angular velocity.