Composition. Under the title Composition and Resolution of Velocity and Forces, we deal with one of the fundamental problems in mechanics—viz. to compound two velocities (or forces) into a single velocity (or force) which shall be their equivalent. We shall consider it as applied to velocities in the first place.
If a point is moving with two independent velocities in any direction, it moves in some one definite direction with a definite speed. This single velocity (for the term includes the idea of direction as well as speed) is equivalent to the two component velocities, and is termed their resultant. A good example is afforded by a ball thrown up in a moving railway carriage; it partakes of the train's motion horizontally, while it also simultaneously moves vertically upwards.
When the two components are in the same straight line, their resultant is in all cases equal to their algebraic sum. In the case of velocities in different directions, the magnitude and direction of their resultant is obtained by the following theorem, known as the Parallelogram of Velocities: If a point A move with two velocities, represented in magnitude and direction by AP and AQ respectively, their resultant will be similarly represented by AR, the diagonal of the parallelogram of which AP and AQ are conterminous sides. For, let the point move along AQ with velocity AQ, and let the page be in motion in the direction AP with velocity AP. After a unit of time has elapsed, the point will have moved from A to Q along AQ, but, owing to the motion of the page, the line AQ will have moved into the position PR, so that the point will really be at R; hence its motion has been in the direction AR, with a velocity whose magnitude is represented by AR.
Similarly, we may compound any number of velocities in one plane into a single resultant. In the case where three components are not coplanar, a corresponding theorem, the Parallelepiped of Velocities, is used to find the resultant.
The resolution of velocities is exactly the converse problem; for where a directed length such as AR can be made the diagonal of a parallelogram, then the conterminous sides are the components. Of course, in this manner, an infinite number of pairs of components can be obtained, each having the given velocity as their resultant. But the resolutions usually required are those in which the components are at right angles.
Since forces can be graphically represented in the same manner as velocities, all that has been said of velocities applies equally well to forces; and obvious changes in the terminology at once give
the means of compounding and resolving forces. See DYNAMICS, KINEMATICS, STATICS.