Addition of Velocities

The spacetime diagram below shows the worldlines of a tree, two cars, and two photons which all go through the origin at O; that is, they all start out from the same place at the same time (in all frames of reference).

When observed from the tree frame, photon 1 is moving to the left with speed c, car 1 is moving to the left with speed c/2, car 2 is moving to the right with speed c/2, and photon 2 is moving to the right with speed c. These speeds can be read off from the spacetime diagram since in the tree frame, when the tree is at I, photon 1 is at G, car 1 is at H, car 2 is at J, and photon 2 is at K. By comparing the separation between the tree at I and photon 1 at G. and the separation between the tree at I and car 1 at H, we can conclude that car 1 is moving at half the speed of light. Similarly, by comparing the separation between the tree at I and photon 2 at K. and the separation between the tree at I and car 2 at J, we can conclude that car 2 is also moving at half the speed of light.

Now let us figure out what the speeds of the tree and car 2 are when observed from the frame of reference fixed to car 1. Answer the following questions:

1. At which spacetime point, labeled A through K, is the tree in the car 1 frame when car 1 is at A?
2. At which spacetime point, labeled A through K, is car 2 in the car 1 frame when car 1 is at A?
3. At which spacetime point, labeled A through K, is photon 2 in the car 1 frame when car 1 is at A?
4. By comparing the separation between car 1 and photon 2, and the separation between car 1 and the tree, figure out the speed of the tree in the car 1 frame.
5. By comparing the separation between car 1 and photon 2, and the separation between car 1 and car 2, figure out the speed of car 2 in the car 1 frame.