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Transforming Velocities

Transforming Velocities. Transforming Velocities. To complete the modification of the Galilean transformations we look at how velocities are transformed. S′. y. y′. x′. x. S. z. z′. Transforming Velocities.

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Transforming Velocities

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  1. Transforming Velocities

  2. Transforming Velocities • To complete the modification of the Galilean transformations we look at how velocities are transformed.

  3. S′ y y′ x′ x S z z′ Transforming Velocities • Consider a particle moving with a velocity as seen by an observer in another inertial frame.

  4. Transforming Velocities As measured in the S frame the components of the velocity are Similarly, in the S′ frame the same vector is

  5. Transforming Velocities Recall the following Lorentz transformations: Therefore we get that, and

  6. Transforming Velocities Thus

  7. Transforming Velocities Similarly for the transform of the y component of the velocity,

  8. And similarly, Transforming Velocities

  9. Transforming Velocities Equivalently we can show that velocity transformations from the primed to the unprimed coordinates are:

  10. Example: Relative Velocities

  11. Example: Relative Velocities Ex: Two spaceships A and B are moving in opposite directions (see fig.). An observer on the earth measures the speed of A to be 0.75c and the speed of B to be 0.85c. Find the velocity of B wrt A.

  12. 0.85c 0.75c y x Earth Frame z Example: Relative Velocities B A

  13. Example: Relative Velocities • The question requires as to determine how the velocity of B looks in the frame of A.

  14. 0.75c 0.85c V′B 0.75c y y′ x x ′ Ship A Frame Earth Frame z z ′ Example: Relative Velocities B B A

  15. The velocity of B as seen from the frame of A Or rather Example: Relative Velocities • Solution:

  16. The velocity of B as seen from the frame of A Or rather Example: Relative Velocities • Solution:

  17. The velocity of B as seen from the frame of A Or rather Example: Relative Velocities • Solution:

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