An observer is staitonary in the inertial system S. A car is moving in the x-dir

Question

An observer is staitonary in the inertial system S. A car is moving in the x-direction with speed u = c/2 with respeect to the car, and thus the driver is in a different inertial frame s'. The passenger of the car takes out a baseball, opens the sunroof of the car and throws out the ball with a velocity v_B=c/2 (a) if the baseball is thrown in the x direction (in the same direction the car is moving) what is the speed of the baseball as seen by the observer is S? (b) If the baseball is thrown in the minus x-direction (opposite to the direction the car is moving), what is the speed of the baseball as seen by the observer in s? (c) If the baseball is thrown in the y-direction (perpendicular to the direction the car is moving), what is the velocity (in direction x, y and z) of the baseball as seen by the observer in S (Give all three velocity components!)

Explanation / Answer

einstein veloicty relationship states that:

if u=veloicty of an object seen by A

u'=veloicty of object seen by B

v=speed of B as seen by A

then u=(v+u')/(1+(v*u/c^2))

now, here :

A is the inertial system S which is at rest

B is the moving car

v=c/sqrt(2)

u'=veloicty of baseball as seen from moving car (i.e. B)=c/sqrt(2)=v

then velocity of baseball seen by A i.e. the stationary observer=(v+u')/(1+(v*u/c^2))

=(v+v)/(1+(v*v/c^2))

=2*v/(2+(v/c)^2))

=(2*c/sqrt(2))/(1+(1/2))

=0.94281*c

part b:

if it is thrown in minus x direction, then u'=-c/sqrt(2)=-v

then speed seen by A=0 (as v+u'=0)

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