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Relativistic Velocity Addition

L ight coming from a headlight of a car traveling at v : It travels at c , not v + c , with respect to a stationary observer!. Relativistic Velocity Addition.  This section will introduce the formula which will “resolve” this “paradox.”.

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Relativistic Velocity Addition

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  1. Light coming from a headlight of a car traveling at v: It travels at c, not v + c,with respect to a stationary observer! Relativistic Velocity Addition This section will introduce the formula which will “resolve” this “paradox.” Consider a fast-traveling jet plane with a wing-mounted air-to-air missile

  2. u’ v Relativistic Addition of Velocities While flying at speed v with respect to the control tower, the Missileis launched, traveling with a speed u’ relative to the plane: Relativity The formula for the speed u of the missile with respect to the “ground” is given by v + u’ u = vu’ c2 1 + 2 v = velocity of object 1 wrt inertial observer 1 u = velocity of object 2 wrt inertial observer u’ = velocity of object 2 wrt object 1

  3. The CLASSICAL RESULT would say that u = v + u’. Thus, the relativistic addition REDUCES the classical result by a factor of 1/[1 + vu’/c2]. If object 2 is moving in the OPPOSITE direction to object 1, then u’ is NEGATIVE.

  4. Velocity Transformation………………………. ---Theoretical formulation--- ……………………( We can use the Lorentz transformation equations to derive the relativistic velocity transformation. We consider only one dimensional motion along the x axis. Use the following definition to obtain the velocity transformation s Velocity transformations Q: when

  5. Q: Using the following definition , obtain the following results for y component of velocities ? Answer

  6. A spaceship traveling at v = .8cwrt earth launches a research satellite in a forward direction so that it moves at .5cwrt the spaceship. How fast is it going wrt earth? First note that classically the satellite would be traveling at .8c + .5c = 1.3c, which we know is not possible. Invoking the relativistic addition of velocities we have v = .8c and u’ = .5c so that v + u’ u = vu’ c2 1 +

  7. First note that classically the satellite would be traveling at .8c + c = 1.8c, which we know is not possible. Invoking the relativistic addition of velocities we have v = .8c and u’ = c so that v + u’ u = vu’ c2 1 +

  8. A spaceship traveling at v = .8cwrt earth turns on a laser light in a backward direction. How fast is the laser light traveling wrt earth? First note that classically the satellite would be traveling at .8c + -c = -.2c, which we know is not possible. Invoking the relativistic addition of velocities we have v = .8c and u’ = -c so that v + u’ u = vu’ c2 1 +

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