where x, x′ are any two corresponding distances; e.g. they may be the initial distances, both particles being supposed to start from rest. The consideration of dimensions was introduced by J. B. Fourier (1822) in connexion with the conduction of heat.
| Fig. 64. |
§ 13. General Motion of a Particle.—Let P, Q be the positions of a moving point at times t, t + δt respectively. A vector OU> drawn parallel to PQ, of length proportional to PQ/δt on any convenient scale, will represent the mean velocity in the interval δt, i.e. a point moving with a constant velocity having the magnitude and direction indicated by this vector would experience the same resultant displacement PQ> in the same time. As δt is indefinitely diminished, the vector OU> will tend to a definite limit OV>; this is adopted as the definition of the velocity of the moving point at the instant t. Obviously OV> is parallel to the tangent to the path at P, and its magnitude is ds/dt, where s is the arc. If we project OV> on the co-ordinate axes (rectangular or oblique) in the usual manner, the projections u, v, w are called the component velocities parallel to the axes. If x, y, z be the co-ordinates of P it is easily proved that
| u = | dx | , v = | dy | , w = | dz | . |
| dt | dt | dt |
(1)
The momentum of a particle is the vector obtained by multiplying the velocity by the mass m. The impulse of a force in any infinitely small interval of time δt is the product of the force into δt; it is to be regarded as a vector. The total impulse in any finite interval of time is the integral of the impulses corresponding to the infinitesimal elements δt into which the interval may be subdivided; the summation of which the integral is the limit is of course to be understood in the vectorial sense.
Newton’s Second Law asserts that change of momentum is equal to the impulse; this is a statement as to equality of vectors and so implies identity of direction as well as of magnitude. If X, Y, Z are the components of force, then considering the changes in an infinitely short time δt we have, by projection on the co-ordinate axes, δ(mu) = Xδt, and so on, or
| m | du | = X, m | dv | = Y, m | dw | = Z. |
| dt | dt | dt |
(2)
For example, the path of a particle projected anyhow under gravity will obviously be confined to the vertical plane through the initial direction of motion. Taking this as the plane xy, with the axis of x drawn horizontally, and that of y vertically upwards, we have X = 0, Y = −mg; so that