As seen in the lab frame a rocket A is moving to the right
with speed c and rocket B is moving to the left with speed
c . The pilot of rocket B fires a laser beam. In the B-frame,
the laser beam points in the x-y plane and makes and angle
=53.13˚ with the x-axis. (It has a positive x- component of
velocity and a positive y- component of velocity).
a) Find the x and y components of velocity of the laser beam
in the B-frame.
b) Show that the speed of the rocket B as seen by the observer in the A-frame is u Kc where
2
2
1
K .
c) Find the x and y components of velocity of the laser beam as seen by the observer in the A-frame in terms of K.
d) Find the angle between the laser beam and the x-axis as measured by the observer in the A-frame in terms of K.
e) For what value of does A see the laser pointing
along the y-direction?
To solve this problem, we'll use the Lorentz transformation equations. These equations allow us to relate the coordinates and velocities measured in two different reference frames moving relative to each other.
Let's start by defining the variables:
- Rocket A is moving to the right with speed cβ in the lab frame.
- Rocket B is moving to the left with speed cβ in the lab frame.
- The laser beam is fired by the pilot of rocket B in the B-frame.
- The laser beam points in the x-y plane in the B-frame, making an angle θ=53.13° with the x-axis.
a) To find the x and y components of velocity of the laser beam in the B-frame, we need to apply the Lorentz transformation equations for velocity. The x and y components of the velocity in the B-frame are given by:
Vx' = (Vx - βc)/(1 - (Vxβ/c^2))
Vy' = Vy/γ
Where Vx and Vy are the x and y components of the velocity in the lab frame, and Vx' and Vy' are the transformed velocities in the B-frame.
b) To show that the speed of rocket B as seen by an observer in the A-frame is given by √(1+β^2)Kc, we need to calculate the velocity of rocket B as seen by an observer in the A-frame. The velocity of rocket B in the A-frame is given by:
VB' = (VB - βc)/(1 - (VBβ/c^2))
where VB is the velocity of rocket B in the lab frame.
c) To find the x and y components of velocity of the laser beam as seen by the observer in the A-frame, we need to apply the Lorentz transformation equations for velocity again. The x and y components of the velocity in the A-frame are given by:
Vx'' = (Vx' - βc)/(1 - (Vx'β/c^2))
Vy'' = (Vy'√(1 - (Vx'β/c^2)^2))/(1 - (Vx'β/c^2))
where Vx' and Vy' are the transformed velocities of the laser beam from part a).
d) To find the angle between the laser beam and the x-axis as measured by the observer in the A-frame, we take the inverse tangent of the y component of velocity divided by the x component of velocity. The angle θ'' is given by:
θ'' = arctan(Vy''/Vx'')
where Vx'' and Vy'' are the transformed velocities of the laser beam from part c).
e) To find the value of β for which rocket A sees the laser pointing along the y-direction, we set the x component of velocity of the laser beam in the A-frame, Vx'', equal to zero and solve for β.