Classical Velocity Addition

For simplicity, we restrict our consideration of velocity addition to one-dimensional motion. Classically, velocities add like regular numbers in one-dimensional motion. (See this figure.) Suppose, for example, a girl is riding in a sled at a speed 1.0 m/s relative to an observer. She throws a snowball first forward, then backward at a speed of 1.5 m/s relative to the sled. We denote direction with plus and minus signs in one dimension; in this example, forward is positive. Let \(v\) be the velocity of the sled relative to the Earth, \(u\) the velocity of the snowball relative to the Earth-bound observer, and \(u\prime \) the velocity of the snowball relative to the sled.

In part a, a man is pulling a sled towards the right with a velocity v equals one point zero meters per second. A girl sitting on the sled facing forward throws a snowball toward a boy on the far right of the picture. The snowball is labeled u primed equals one point five meters per second in the direction the sled is being pulled. The boy is labelled two point five meters per second. In figure b, a similar figure is shown, but the man’s velocity is one point zero meters per second, the girl is facing backward and throwing the snowball behind the sled. The snowball is labelled u primed equals negative one point five meters per second, and the boy is labelled u equals negative zero point five meters per second.

Classically, velocities add like ordinary numbers in one-dimensional motion. Here the girl throws a snowball forward and then backward from a sled. The velocity of the sled relative to the Earth is \(\text{v=}1\text{.}0\phantom{\rule{0.25em}{0ex}}\text{m/s}\). The velocity of the snowball relative to the sled is \(u\prime \), while its velocity relative to the Earth is \(u\). Classically, vu\(\text{u=v+u}\prime \).

Classical Velocity Addition

\(\text{u=v+u}\prime \)

Thus, when the girl throws the snowball forward, \(u=1.0 m/s+1.5 m/s=2.5 m/s\). It makes good intuitive sense that the snowball will head towards the Earth-bound observer faster, because it is thrown forward from a moving vehicle. When the girl throws the snowball backward, \(u=1.0 m/s+(-1.5 m/s)=-0.5 m/s\). The minus sign means the snowball moves away from the Earth-bound observer.

This lesson is part of:

Special Relativity

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