The time-integral on the right hand is called the impulse of the force on the interval t′ − t. The statement that the increase of momentum is equal to the impulse is (it maybe remarked) equivalent to Newton’s own formulation of his Second Law. The form (1) is deduced from it by putting t′ − t = δt, and taking δt to be infinitely small. In problems of impact we have to deal with cases of practically instantaneous impulse, where a very great and rapidly varying force produces an appreciable change of momentum in an exceedingly minute interval of time.
In the case of a constant force, the acceleration u̇ or ẍ is, according to (1), constant, and we have
| d2x | = α, |
| dt2 |
(3)
say, the general solution of which is
x = 1⁄2 αt2 + At + B.
(4)
The “arbitrary constants” A, B enable us to represent the circumstances of any particular case; thus if the velocity ẋ and the position x be given for any one value of t, we have two conditions to determine A, B. The curve of positions corresponding to (4) is a parabola, and that of velocities is a straight line. We may take it as an experimental result, although the best evidence is indirect, that a particle falling freely under gravity experiences a constant acceleration which at the same place is the same for all bodies. This acceleration is denoted by g; its value at Greenwich is about 981 centimetre-second units, or 32.2 feet per second. It increases somewhat with the latitude, the extreme variation from the equator to the pole being about 1⁄2%. We infer that on our reckoning the force of gravity on a mass m is to be measured by mg, the momentum produced per second when this force acts alone. Since this is proportional to the mass, the relative masses to be attributed to various bodies can be determined practically by means of the balance. We learn also that on account of the variation of g with the locality a gravitational system of force-measurement is inapplicable when more than a moderate degree of accuracy is desired.
We take next the case of a particle attracted towards a fixed point O in the line of motion with a force varying as the distance from that point. If μ be the acceleration at unit distance, the equation of motion becomes
| d2x | = −μx, |
| dt2 |