# Special Relativity

## Relativistic Addition of Velocities


### Learning Objectives

By the end of this section, you will be able to:
1. Calculate relativistic velocity addition.
2. Explain when relativistic velocity addition should be used instead of classical addition of velocities.
3. Calculate relativistic Doppler shift.
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 .) 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  be the velocity of the sled relative to the Earth,  the velocity of the snowball relative to the Earth-bound observer, and  the velocity of the snowball relative to the sled.
Thus, when the girl throws the snowball forward, . 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, . 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 . 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 . But we know that light will move away from the car at speed  relative to the driver of the car, and light will move towards the observer on the sidewalk at speed , too.
Velocities cannot add to greater than the speed of light, provided that  is less than  and  does not exceed . The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.

### Doppler Shift

Although the speed of light does not change with relative velocity, the frequencies and wavelengths of light do. First discussed for sound waves, a Doppler shift occurs in any wave when there is relative motion between source and observer.
In the Doppler equation,  is the observed wavelength,  is the source wavelength, and  is the relative velocity of the source to the observer. The velocity  is positive for motion away from an observer and negative for motion toward an observer. In terms of source frequency and observed frequency, this equation can be written
Notice that the – and + signs are different than in the wavelength equation.
The relativistic Doppler shift is easy to observe. This equation has everyday applications ranging from Doppler-shifted radar velocity measurements of transportation to Doppler-radar storm monitoring. In astronomical observations, the relativistic Doppler shift provides velocity information such as the motion and distance of stars.

### Test Prep for AP Courses


### Section Summary

1. With classical velocity addition, velocities add like regular numbers in one-dimensional motion: , where  is the velocity between two observers,

 is the velocity of an object relative to one observer, and  is the velocity relative to the other observer.
2. Velocities cannot add to be greater than the speed of light. Relativistic velocity addition describes the velocities of an object moving at a relativistic speed:

3. An observer of electromagnetic radiation sees relativistic Doppler effects if the source of the radiation is moving relative to the observer. The wavelength of the radiation is longer (called a red shift) than that emitted by the source when the source moves away from the observer and shorter (called a blue shift) when the source moves toward the observer. The shifted wavelength is described by the equation
     is the observed wavelength,  is the source wavelength, and  is the relative velocity of the source to the observer.

### Conceptual Questions


### Problems & Exercises
