Page images
PDF
EPUB

thus measuring change of direction, and the acceleration of Resolution actual velocity along the curve.

and composition of accelerations.

We may take another mode of resolving acceleration for a plane curve, which is sometimes useful; along, and perpendicular to, the radius-vector. By a method similar to that employed in

+

§ 27, we easily find for the component along the radius-vector

d2r

dt2

d2x
di2

[blocks in formation]
[ocr errors]
[ocr errors]
[ocr errors]

r dt

do 2

dt

موم

tion of the

given velo

celeration.

33. If for any case of motion of a point we have given the Determinawhole velocity and its direction, or simply the components of motion from the velocity in three rectangular directions, at any time, or, as city or acis most commonly the case, for any position, the determination of the form of the path described, and of other circumstances of the motion, is a question of pure mathematics, and in all cases is capable, if not of an exact solution, at all events of a solution to any degree of approximation that may be desired.

The same is true if the total acceleration and its direction at every instant, or simply its rectangular components, be given, provided the velocity and direction of motion, as well as the position, of the point at any one instant, be given.

For we have in the first case

dx
dt

[ocr errors]

three simultaneous equations which can contain only x, y, z, and
t, and which therefore suffice when integrated to determine x, y,
and z in terms of t. By eliminating t among these equations, we
obtain two equations among x, y, and z—each of which repre-
sents a surface on which lies the path described, and whose
intersection therefore completely determines it.

In the second case we have

= u = q cos a, etc.,

[ocr errors][merged small][ocr errors]

to which equations the same remarks apply, except that here
each has to be twice integrated.

Determination of the motion from given velocity or acceleration.

Examples of velocity.

Examples of
accelera-
tion.

The arbitrary constants introduced by integration are determined at once if we know the co-ordinates, and the components of the velocity, of the point at a given epoch.

34. From the principles already laid down, a great many interesting results may be deduced, of which we enunciate a few of the most important.

a. If the velocity of a moving point be uniform, and if its direction revolve uniformly in a plane, the path described is a circle.

Let a be the velocity, and a the angle through which its direction turns in unit of time; then, by properly choosing the axes, we have

[merged small][merged small][ocr errors][merged small][merged small]

dx

dt

dy dt

= a cos at,

d3y

dt2

- му,

b. If a point moves in a plane, and if its component velocity parallel to each of two rectangular axes is proportional to its distance from that axis, the path is an ellipse or hyperbola whose principal diameters coincide with those axes; and the acceleration is directed to or from the origin at every instant.

a2

2

[ocr errors]

a

dy dt

uvy, and the whole acceleration is

d2x
Hence
= μνα,
dt2
towards or from 0.

"

dy V X
Also
dx му
bola referred to its principal axes. (Compare § 65.)

from which μy2 – vx2 = C, an ellipse or hyper

C. When the velocity is uniform, but in direction revolving uniformly in a right circular cone, the motion of the point is in a circular helix whose axis is parallel to that of the cone.

35. a. When a point moves uniformly in a circle of radius K, with velocity V, the whole acceleration is directed towards

the centre, and has the constant value

See § 31.

[ocr errors]

b. With uniform acceleration in the direction of motion, a Examples of point describes spaces proportional to the squares of the times tion. elapsed since the commencement of the motion.

In this case the space described in any interval is that which would be described in the same time by a point moving uniformly with a velocity equal to that at the middle of the interval. In other words, the average velocity (when the acceleration is uniform) is, during any interval, the arithmetical mean of the initial and final velocities. This is the case of a stone falling vertically.

[blocks in formation]

vacuum.

a, therefore

dv

And we may write the equation (§ 29) v

dx

Then

= v at, and x = {ať3.

v = V + at, x =

If at time t = 0 the velocity was V, these equations become at

once

v2

2

And initial velocity = V,
final
Arithmetical mean

x = Vt + ať2, and

d2z
dt2

[ocr errors]

dz

dt

[ocr errors]

= √ + at;

= a, whence

[ocr errors]

V2

2

√ + sat,

[ocr errors]

= a α;

whence the second part of the above statement.

c. 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

+ ax.

For if the axis of y be parallel to the acceleration a, and if the plane of xy be that of motion at any time,

0,

0, z=0,

v2

2

and therefore the motion is wholly in the plane of xy.

d2x

d'y

0,

dt2

dt2

= ax.

Examples of acceleration.

and by integration

x= Ut+a, y = {ať2 + Vt + b,

X =

where U, V, a, b are constants.

The elimination of t gives the equation of a parabola of which the axis is parallel to y, parameter and vertex the point whose co

202

a

ordinates are

a

UV

V2

ω

tortuosity V v2 + y22 w2

[merged small][ocr errors]

Tortuosity =

The curvature of the path is

d. As an illustration of acceleration in a tortuous curve, we take the case of § 13, or of § 34, c.

tan a =

[ocr errors]

Let a point move in a circle of radius r with uniform angular velocity (about the centre), and let this circle move perpendicular to its plane with velocity V. The point describes a helix on a cylinder of radius r, and the inclination a is given by V

y = b

A

V2 + Ar

γω

Vw

V2 + r2 w2 •

1 p2w2

r V 2 + r2 w

v2

2a

V

A

√ Ár V2 + Ar

The acceleration is rw3, directed perpendicularly towards the axis of the cylinder.-Call this A.

Curvature=

A

2

√2 +

[ocr errors]

A2

ω

[ocr errors]

or

2

Vw

[ocr errors]

A

Curvature (in the limit) =

V2

A2

rw2
V2 + r2 w 2

2

ω

[ocr errors]

2

ω

Let A be finite, indefinitely small, and therefore o indefinitely great.

and the

Tortuosity (

Thus, if we have a material particle moving in the manner specified, and if we consider the force (see Chap. II.) required to produce the acceleration, we find that a finite force perpendicular to

of accelera

the line of motion, in a direction revolving with an infinitely Examples great angular velocity, maintains constant infinitely small de- tion. flection (in a direction opposite to its own) from the line of undisturbed motion, finite curvature, and infinite tortuosity.

As in case c,

e. When the acceleration is perpendicular to a given plane and proportional to the distance from it, the path is a plane curve, which is the harmonic curve if the acceleration be towards the plane, and a more or less fore-shortened catenary (§ 580) if from the plane.

perpendicular to
at any instant.

Hence

and

d2z
dt2

y = P cos

=

0,

0, and 20, if the axis of z be

dz
dt
the acceleration and to the direction of motion
Also, if we choose the origin in the plane,

d2y

0,

му.

dť2

d2x

dt2

[merged small][ocr errors][merged small][merged small]

This gives, if u is negative,

μ

[ocr errors]

(~+Q), the harmonic curve, or curve of sines.

X

If μ be positive,

b

y = På + Qe ̃ ̄† ;

Q€

and by shifting the origin along the axis of x this can be put in
the form

[merged small][ocr errors]

which is the catenary if 2R = b; otherwise it is the catenary
stretched or fore-shortened in the direction of y.

directed to a

36. [Compare §§ 233-236 below.] a. When the accele- Acceleration ration is directed to a fixed point, the path is in a plane passing fixed centre. through that point; and in this plane the areas traced out by the radius-vector are proportional to the times employed. This includes the case of a satellite or planet revolving about its primary.

Evidently there is no acceleration perpendicular to the plane containing the fixed and moving points and the direction

« PreviousContinue »