The Composition of Velocities
Table of Contents
In the system k moving along the axis of X of the system K with velocity v, let a point move in accordance with the equations
ξ = wξ τ, η = wη τ, ζ = 0,
where wξ and wη denote constants.
Required: the motion of the point relatively to the system K. If with the help of the equations of transformation developed in § 3 we introduce the quantities x, y, z, t into the equations of motion of the point, we obtain
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Thus the law of the parallelogram of velocities is valid according to our theory only to a first approximation. We set
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a is then to be looked upon as the angle between the velocities v and w. After a simple calculation we obtain
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7 Not a pendulum-clock, which is physically a system to which the Earth belongs. This case had to be excluded.
11 It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. If w also has the direction of the axis of X, we get
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It follows from this equation that from a composition of two velocities which are less than c, there always results a velocity less than c. For if we set v = c − κ, w = c − λ, κ and λ being positive and less than c, then
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It follows, further, that the velocity of light c cannot be altered by compo- sition with a velocity less than that of light. For this case we obtain
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We might also have obtained the formula for V, for the case when v and w have the same direction, by compounding two transformations in accordance with §
- If in addition to the systems K and k figuring in § 3 we introduce still another system of co-ordinates k 0 moving parallel to k, its initial point moving on the axis of X with the velocity w, we obtain equations between the quantities x, y, z, t and the corresponding quantities of k 0 , which differ from the equations found in § 3 only in that the place of “v” is taken by the quantity
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from which we see that such parallel transformations—necessarily—form a group.
We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.