Relativistic addition of velocities
PROBLEM SET
PROBLEM 1. Show that the relativistic addition of velocities formulas
reduce to the Newtonian ones whenever v << c.
PROBLEM 2.
Show that the equations
(dx - vdt)
dx'
=
,
dt' (dt - dxv/c2)
dy'
dz'
2
2
dt' = dy/(dt - dxv/c ), dt' = dz/(dt - dxv/c ),
in fact do clean up to
Vx - v
Vx' = 1-vVx/c2 ,
Vy' =
Vy(1-v2/c2)1/2
,
1-vVx/c2
Vz' =
Vz(1-v2/c2)1/2
.
1-vVx/c2
PROBLEM 3. Suppose (S) is an IRF, and a particle is detected in (S) to
have a speed Vy = c. Suppose (S') is another IRF moving with constant
velocity v with respect to (S) in the positive x-direction.
Use the
relativistic addition of velocities and (V')2 = (Vx')2 + (Vy')2 + (Vz')2 to
prove that V' = c. Hint: This time Vy'  0.
PROBLEM 4. Suppose (S) is an IRF, and a particle is detected in (S) to
have a speed Vz = c. Suppose (S') is another IRF moving with constant
velocity v with respect to (S) in the positive x-direction.
Use the
relativistic addition of velocities and (V')2 = (Vx')2 + (Vy')2 + (Vz')2 to
prove that V' = c. Hint: This time Vz'  0.
PROBLEM 5. Suppose you are in (S') which is moving at a speed of c
relative to (S) in the direction of the positive x-axis. Suppose that
in (S), a particle is moving at the speed of light in the negative xdirection. What is its speed in your frame? Hint: If you substitute
Vx = v = c you get 0/0, an indeterminate form. If you try L'Hospital's
rule, you still have an indeterminate form. Try substituting Vx = c and
simplifying before substituting v = c.
PROBLEM 6. Suppose
relative to (S) in
in (S), a particle
direction. What is
you are in (S') which is moving at a speed of c
the direction of the positive x-axis. Suppose that
is moving at the speed of light in the positive xits speed in your frame?
PROBLEM 7. Suppose (S) is an IRF, and a particle is detected in (S) to
have a speed V = (c/22)(i + j).
(a)
What is the particle's speed in (S)?
Hint: V2 = (Vx)2 + (Vy)2 +
(Vz)2
(b)
What is the particle's speed in (S') (another IRF moving with
constant velocity v with respect to (S) in the positive xdirection). Prove it.