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Curve of Pursuit. The idea is said to have been suggested Curves of by the old rule of steering a privateer always directly for the vessel pursued. (Bouguer, Mém. de l'Acad. 1732.) It is the curve described by a dog running to its master.
The simplest cases are of course those in which the first point moves in a straight line, and of these there are three, for the velocity of the first point may be greater than, equal to, or less than, that of the second. The figures in the text below represent the curves in these cases, the velocities of the pursuer being, 1, and of those of the pursued, respectively. In the second and third cases the second point can never overtake the first, and consequently the line of motion of the first is an asymptote. In the first case the second point overtakes the first, and the curve at that point touches the line of motion of the first. The remainder of the curve satisfies a modified form of statement of the original question, and is called the Curve of Flight.
Curves of pursuit.
We will merely form the differential equation of the curve, and give its integrated form, leaving the work to the student.
Suppose Ox to be the line of motion of the first point, whose velocity is v, AP the curve of pursuit, in which the velocity is u, then the tangent at P always passes through Q, the instantaneous position of the first point. It will be evident, on a moment's consideration, that the curve AP must have a tangent
perpendicular to Ox. Take this as the axis of y, and let OA =α. Then, if OQ = &, AP = s, and if x, y be the coordinates of P, we have
because A, O and P, Q are pairs of simultaneous positions of the two points.
- s = es = x Y
From this we find, unless e = 1,
2 (2 + 2) = _***
you1 (e − 1 ) ;
a2 (e + 1) ye
and if e = 1,
the only case in which we do not get an algebraic curve.
41. 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 as 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 instant 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.
Unit of angular velocity is that of a point which describes, Angular 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 to the radius; being an angle of
= 57°. 29578 ... = 57° 17′ 44′′.8 nearly.
For brevity we shall call this angle a radian.
42. The rate of increase or diminution of the angular velo- Angular accity when variable is called the angular acceleration, and is measured in the same way and by the same unit.
By methods precisely similar to those employed for linear velocity and acceleration we see that if ◊ be the angle-vector of a point moving in a plane-the
Angular velocity is w=
Angular acceleration is
Since (§ 27) r
is the velocity perpendicular to the radius
vector, we see that
The angular velocity of a point in a plane is found by dividing the velocity perpendicular to the radius-vector by the length of the radius-vector.
43. When one point describes uniformly a circle about Angular another, the time of describing a complete circumference being T, we have the angle 2π described uniformly in T; and, there
Even when the angular velo
fore, the angular velocity is
When a point moves uniformly in a straight line its angular velocity evidently diminishes as it recedes from the point about which the angles are measured.
The polar equation of a straight line is
r = a sec 0.
But the length of the line between the limiting angles 0 and 0 is a tan 6, and this increases with uniform velocity v. Hence
(a tan 0) = a sec* ◊
and is therefore inversely as the square of the
Similarly for the angular acceleration, we have by a second. differentiation,
(1 - 0) 14,
the third power of the radius-vector.
and ultimately varies inversely as
44. We may also talk of the angular velocity of a moving
of a plane. 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 definite information. This will be supplied in a subsequent section.
45. All motion that we are, or can be, acquainted with, is Relative merely. We can calculate from astronomical data for any instant the direction in which, and the velocity with which we are moving on account of the earth's diurnal rotation. We may compound this with the similarly calculable velocity of the earth in its orbit. This resultant again we may compound with the (roughly known) velocity of the sun relatively to the so-called fixed stars; but, even if all these elements were accurately known, it could not be said that we had attained any idea of an absolute velocity; for it is only the sun's relative motion among the stars that we can observe; and, in all probability, sun and stars are moving on (possibly with very great rapidity) relatively to other bodies in space. We must therefore consider how, from the actual motions of a set of points, we may find their relative motions with regard to any one of them;
and how, having given the relative motions of all but one with Relative regard to the latter, and the actual motion of the latter, we may find the actual motions of all. The question is very easily answered. Consider for a moment a number of passengers walking on the deck of a steamer. Their relative motions with regard to the deck are what we immediately observe, but if we compound with these the velocity of the steamer itself we get evidently their actual motion relatively to the earth. Again, in order to get the relative motion of all with regard to the deck, we abstract our ideas from the motion of the steamer altogether-that is, we alter the velocity of each by compounding it with the actual velocity of the vessel taken in a reversed direction.
Hence to find the relative motions of any set of points with regard to one of their number, imagine, impressed upon each in composition with its own velocity, a velocity equal and opposite to the velocity of that one; it will be reduced to rest, and the motions of the others will be the same with regard to it as before.
Thus, to take a very simple example, two trains are running in opposite directions, say north and south, one with a velocity of fifty, the other of thirty, miles an hour. The relative velocity of the second with regard to the first is to be found by impressing on both a southward velocity of fifty miles an hour; the effect of this being to bring the first to rest, and to give the second a southward velocity of eighty miles an hour, which is the required relative motion.
Or, given one train moving north at the rate of thirty miles. an hour, and another moving west at the rate of forty miles an hour. The motion of the second relatively to the first is at the rate of fifty miles an hour, in a south-westerly direction inclined to the due west direction at an angle of tan13. It is needless to multiply such examples, as they must occur to every one.
46. Exactly the same remarks apply to relative as compared with absolute acceleration, as indeed we may see at once, since accelerations are in all cases resolved and compounded by the same law as velocities.