## Sunday, 5 October 2014

### Relative velocities

Let's say you're standing on an interstate freeway that runs north and south. You see a truck heading north at 60 km/h, and a car heading south at 70 km/h. All three of you (you, the car driver, and the truck driver) agree on these points:
• the truck is traveling at 60 km/h north relative to you, and 130 km/h north relative to the car.
• the car is traveling at 70 km/h south relative to you, and 130 km/h south relative to the truck.
• you are traveling at 60 km/h south relative to the truck, and 70 km/h north relative to the car.
Two observers always agree on their relative velocity. The simple addition we used to get the velocity of the truck relative to the car can not be applied to a relativistic situation, however.

Let's say you now stand on an intergalactic freeway. You see a truck heading in one direction at 0.6c, and a car heading in the opposite direction at 0.7c. What is the velocity of the truck relative to the car? It is not 1.3c, because nothing can travel faster than c. The relative velocity can be found using this equation:

In this case, u is the velocity of the car relative to you, v is the velocity of the truck relative to you, and u' is the velocity of the car relative to the truck. Taking the direction the car is traveling to be the positive direction:

So, now everyone involved agrees on this:
• the truck is traveling at 0.6c relative to you, and 0.915c relative to the car.
• the car is traveling at 0.7c relative to you, and 0.915c relative to the truck.
• you are traveling at 0.6c relative to the truck, and 0.7c relative to the car.
The relativistic equation for velocity addition shown above can also be used for non-relativistic velocities. We're more used to adding velocities like this : u' = u - v. This is exactly what the relativistic equation reduces to for velocities much less than the speed of light. The relativistic equation applies to any situation; the one we're used to is a special case that applies only for small velocities.