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  • Calculate relativistic velocity addition.
  • Explain when relativistic velocity addition should be used instead of classical addition of velocities.
  • Calculate relativistic Doppler shift.
A man with oar in his hand is kayaking downstream in a shallow fast-flowing river.
The total velocity of a kayak, like this one on the Deerfield River in Massachusetts, is its velocity relative to the water as well as the water’s velocity relative to the riverbank. (credit: abkfenris, Flickr)

If you’ve ever seen a kayak move down a fast-moving river, you know that remaining in the same place would be hard. The river current pulls the kayak along. Pushing the oars back against the water can move the kayak forward in the water, but that only accounts for part of the velocity. The kayak’s motion is an example of classical addition of velocities. In classical physics, velocities add as vectors. The kayak’s velocity is the vector sum of its velocity relative to the water and the water’s velocity relative to the riverbank.

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 [link] .) 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 size 12{v} {} be the velocity of the sled relative to the Earth, u size 12{u} {} the velocity of the snowball relative to the Earth-bound observer, and u size 12{u rSup { size 8{'} } } {} 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 v= 1 . 0 m/s size 12{ ital "v="1 "." 0`"m/s"} {} . The velocity of the snowball relative to the truck is u size 12{u rSup { size 8{'} } } {} , while its velocity relative to the Earth is u size 12{u} {} . Classically, u=v+u .

Classical velocity addition


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.

Relativistic velocity addition

The second postulate of relativity (verified by extensive experimental observation) says that classical velocity addition does not apply to light. Imagine a car traveling at night along a straight road, as in [link] . If classical velocity addition applied to light, then the light from the car’s headlights would approach the observer on the sidewalk at a speed u=v+c size 12{ ital "u=v+c"} {} . But we know that light will move away from the car at speed c size 12{c} {} relative to the driver of the car, and light will move towards the observer on the sidewalk at speed c size 12{c} {} , too.

A car is moving towards right with velocity v. A boy standing on the side-walk observes the car. The velocity of light u primed is shown to be c as observed by the girl in the car and the velocity of light u is also c as observed by the boy.
According to experiment and the second postulate of relativity, light from the car’s headlights moves away from the car at speed c size 12{c} {} and towards the observer on the sidewalk at speed c size 12{c} {} . Classical velocity addition is not valid.

Questions & Answers

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Practice Key Terms 3

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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