Velocity addition

Problem:

For two inertial coordinate systems K and K' in relative motion at speed “v” along their x1 axes, the Lorentz transformations provide that
x1' = γ(x1 – vt)
x2' = x2
x3' = x3 and
t' = γ(t – vx1/c2), where γ = [1 – v2/c2] .
(a)  Derive the velocity transformations relating speeds uj (in K) and uj' (in K'), with j = 1, 2, 3.
(b)  Now assume Kurt (in system K) fires an energetic photon straight up, along his x2 (or y) axis.  If Paula's system K' travels at v = 0.6 c relative to Kurt, calculate the components of the photon's velocity in Paula's reference frame.
(c)   From your results in part (b), calculate the magnitude of the photon's velocity in Paula's frame.  Explain why this result is to be expected.

Solution:

Problem:

A rocket is traveling towards Earth at 0.6 c.  It fires a projectile of mass M with a velocity of 0.8 c in the rocket frame towards Earth.  
(a)  What is the energy of the projectile in the Earth frame?
(b)  Another rocket is traveling at 0.9 c perpendicular to a line from the rocket to Earth and fires a projectile at 0.8 c in the rocket frame and at 90 degrees from the direction of the rocket's motion (in the rocket frame), what is the energy of the projectile in the Earth frame?
(c)  What is the tangent of the angle of the projectile relative to the ship's motion in the Earth frame?

Solution:

Problem:

A pion, moving along x-axis in the positive x-direction with β = 0.8 in the lab system decays by emitting a muon with β*  = 0.268 in its own rest frame.
(a)  Find the velocity of the muon (magnitude and direction) in the lab frame if the muon is emitted along the incident direction in the rest system of pion.
(b)  Now assume the muon is emitted along the y-axis in the rest frame of the pion.  Find the velocity of muon in the lab frame.
(c)  Now assume the muon is emitted along the positive y-axis (i.e. perpendicular to the incidental direction of pion in the lab frame).  Find the speed of muon in the lab frame and the direction of emission in the rest frame of pion.  For this part assume β = 0.2

Solution:

Problem:

In the reference frame of an outside observer two particles move towards each other, both with relativistic speed v . The angle between them is 2θ as shown in the figure below.  What is the speed of one of the particles as viewed by the other?

image

Solution:

Problem:

Two spaceships, A and B, are moving along a line in opposite directions.  An observer on Earth measures the speed of spaceship A to be 2*108 m/s and the speed of B to be 1*108 m/s.  
When the captain of spaceship A receives a collision warning, the two ships are separated by 3*109 m according to spaceship A's measurement   How long does spaceship A's captain have to avoid a collision?

Solution:

Problem:

A spaceship whose rest length is 350 m has a speed of 0.82 c in a certain reference frame.  A micrometeorite, also with a speed of 0.82 c in this same frame, passes the spaceship on an anti-parallel track.  According to an astronaut in the spaceship, how long does it take the micrometeorite to traverse the entire length of the spaceship?

Solution:

Problem:

In reference frame K particle moves with constant velocity u from a source at the origin to a detector located at x = y = z = 10 m in 10-7 s.
Reference frame K' moves with respect to K with velocity v = 2*108 m/s i.
(a)  What is the speed of the particle in K?
(b)  What is the separation between the source and the detector in K'?
(c)  What is the speed of the particle in K'?

Solution:

Problem:

A space ship is moving to the east at a speed of 0.9c relative to the earth.  A second spaceship is moving to the west at a speed of 0.8c relative to the earth.  What is the speed of one spaceship relative to the other?

Solution:

Problem:

A person on Earth observes two rocket ships moving directly toward each other and colliding.  At time t = 0 in the Earth frame, the Earth observer determines that rocket 1, traveling to the right at v1 = 0.8c, is at point a, and rocket 2 is at point b, traveling to the left at v2 = 0.6c.  They are separated by a distance 4.2*108 m.
(a)  In the Earth frame, how much time will pass before the rockets collide?
(b)  How fast is rocket 2 approaching in rocket 1's frame?  How fast is rocket 1 approaching in rocket 2's frame?
(c)  How much time will elapse in 1's frame from the time rocket 1 passes point a until collision?  How much time will elapse in 2's frame from the time rocket 2 passes point b until collision?

Solution:

Problem:

A train car of length L' = L0 in the train's frame of reference moves along a track at velocity vT = c/2.
A gunman in the rear of car fires a bullet in the direction of motion of the train. 
He observes its velocity to be vB' = c/2.  For an observer in the rest frame of the track, what are the values of
(a)  L, the length of the car?
(b)  vB, the velocity of the bullet?
(c)  t2, the length of time the bullet travels before striking the front of the car?
(d)  x2, the distance the bullet travels?

 Solution: