Let EF be the moving line, and, while the point P moves from E to F, let the line move from AB to CD. Then PE is to EA as the velocity of the point along the line is to the velocity of the line, that is, as CD to AC;

Therefore APD is a straight line, and the point P moves uniformly from A to D.

AD therefore represents the resultant velocity in magnitude and direction.

This proposition is called the Parallelogram of velocities.

In the same way, if a point have two coexistent accelerations, taking AB and AC to represent the velocities added per unit of time, or which would be added per unit of time, due to the accelerations at the instant in question, it follows that AD represents the resultant velocity superposed, or which would be superposed, per unit of time.

This is the Parallelogram of accelerations.

Conversely, any velocity or acceleration, represented by a line AD, can be decomposed into two velocities or accelerations, AB, AC, in any assigned directions.

6. Change of units in the measures of velocities and accelerations.

If v be the measure of a velocity, the meaning is that v units of length are passed over in the unit of time.

If a feet and t seconds be the units, and if v be the

measure of the same velocity when a' feet and t' seconds are units, it follows that

for each expression represents the velocity in feet per second.

If ƒ be the measure of an acceleration when a feet and

t seconds are units, the meaning is that

the velocity per t seconds added in t seconds=fa in feet;

If f' be the measure of the same acceleration when a'

f'a'

feet and t' seconds are units, it follows that fa

t's

is the measure

of the same acceleration referred to a foot and a second, and

fa_f'a'
t2 t's

7. Angular velocity and angular acceleration.

If a straight line turn round in a plane it is said to have angular velocity, and if this angular velocity be variable it is said to have angular acceleration.

If be the inclination, at the time t, of the moving line to any fixed line in the plane, then, exactly as in Articles (3) and (4), the angular velocity is or , and the angular

de

dt

It must be observed that this is quite independent of any motion of translation which the line may have, and simply measures the rate of turning round.

When we speak of the angular velocity of a point P, moving in a plane, about a fixed point 0 in the plane, we really mean the angular velocity of the straight line OP.

If the velocity and direction of motion of P be given, a simple expression can be obtained for its angular velocity about a fixed point 0.

For if OP=r, and if p be the perpendicular from 0 on the direction of motion of the point, and v its velocity, the angular velocity is equal to the resolved part of the velocity perpendicular to OP divided by OP, and therefore

An expression may also be given for the angular velocity of the direction of motion.

If be the inclination to any fixed line in the plane of motion of the normal to the path of the moving point, the angular velocity of the tangent at the point

where p is the radius of curvature, at the point, of the path.

8. Expressions for accelerations.

If x, y are the coordinates, at the time t, of a point moving in a plane, referred to a pair of fixed rectangular axes in the plane, x and y are the distances of the point from the axes of y and x.

The velocity parallel to x is the rate of increase of the distance from the axis of y, and, as in Art. (3), is represented dx by or by i, employing fluxional notation.

dt

dy dt

Similarly or y, is the expression for the velocity parallel to the axis of y.

If these velocities are u and v, the accelerations parallel to du dv

the axes are, by the same reasoning as in Art. (4), and

or ù and v; that is, they are

d2x and day dta

dt dt'

dt'

or ï and ÿ.

We shall in all cases limit the use of the symbols å, ï to the case in which the time is the independent variable.

9. It must be very carefully observed that the velocity of a point in any direction is the rate of change of the distance in that direction, and is equal to the limit of the change of distance divided by the change of time, when that change is indefinitely small.

And similarly, the acceleration in any direction is the rate of change of the velocity in that direction, and is equal to the limit of the change of velocity divided by the change of time, when that change is indefinitely small.

10. Radial and transversal velocities and accelerations.

P

P

Let r, be the polar co-ordinates of a moving point, and u, v the radial and transversal velocities, that is, the velocities in direction of OP and perpendicular to OP.

P being the position of the point at the time t, and P' at the time t + St, and if OP' =r+dr,

If u + Su, v + dv, be the velocities at P' in direction of and perpendicular to OP', acceleration in direction OP

acceleration perpendicular to OP

= limit of (v +dv) cos 80 + (u + du) sin 80 – v

δε

It should be noticed that, if r = 0, that is, if the moving point is passing through the origin, these expressions are

+ and 2r0.

11. The expressions for radial and transversal accelerations may otherwise be obtained in the following manner. Since x = r cos 0, and y=r sin 0, we obtain

=(i — r0) cos 0 — (rÜ + 2r0) sin 0,

12. Case of uniform motion in a circle.

If r and are both constant, and if Ø=w, the transversal acceleration vanishes, and the radial acceleration that is, the resultant acceleration is directed to the centre of

=- - w2r;