Muons decay with a half-life of t½ = 1.56 μs. They are produced with very high speeds in the upper atmosphere when
cosmic rays collide with air molecules. Assume the height of the atmosphere of
Earth is 200 km in the reference frame of the earth.
(a) If the muons are traveling at 0.99998c, how long does it take for the
muons to pass through Earth's atmosphere as viewed from the Earth?
(b) How many half-lives is this?
(c) How long does it take as viewed from the rest frame of the muons?
(d) What is the height of the atmosphere in the rest frame of the muons?
Solution:
Consider two particles with mass m. The first particle, with energy E, collides
elastically with the second particle, which is at rest.
(a) If both scatter at an angle θ with respect to the incoming particle's
momentum, write down the value of θ in terms of E and m.
(b) What value does θ approach in the relativistic limit?
(c) What value does θ approach in the non-relativistic limit?
Solution:
A horizontal one-meter-long stick is moving with velocity u j along the y−direction in the lab frame S. Reference frame S' is moving with velocity v = v i along the x-direction relative to S.
(a) Using the Lorentz transformation, derive the velocity u' of the
stick measured by an observer at rest in S' in terms of u, v, and c.
(b) In the S frame the stick is oriented horizontally. Assume that the stick
is crossing the x-axis at t = 0, with its left and right ends at xL =
−0.5 m, yL = 0 and xR = +0.5 m, yR = 0
respectively.
The rod is not horizontal for observers in the S' frame. Find the vertical
distance that the rod travels while crossing the x'-axis according to an
observer at rest in S', in terms of u, v and c.
(c) Find the length of the stick and the angle θ that the stick makes with the
x'-axis in terms of u, v, and c according to the observer in the S' frame.
Solution:
We can also proceed by transforming the coordinates of events. Let us look
at the positions of the left and the right end of the stick at t = 0 and at t =
t1 in S and S'.
To transform the coordinates of an event from S to S' b we
use the matrix equation below.
x0' | ||
x' | ||
y' | ||
z' |
=
γ | -γβ | 0 | 0 | ||
-γβ | γ | 0 | 0 | ||
0 | 0 | 1 | 0 | ||
0 | 0 | 0 | 1 |
x0 | ||
x | ||
y | ||
z |
.
Event: | coordinates in S | coordinates in S' |
(1) left edge at t = 0 | (0, -0.5, 0, 0) | (0.5γβ, -0.5γ , 0, 0) |
(2) right edge at t = 0 | (0, 0.5, 0, 0) | (-0.5γβ, 0.5γ, 0, 0) |
(3) left edge at t = t | (ct, -0.5, ut, 0) | (γct+0.5γβ, -γβct-0.5γ, ut, 0) |
(4) right edge at t = t | (ct, 0.5, ut, 0) | (γct-0.5γβ, -γβct+0.5γ, ut, 0) |
(All quantities are in SI units.)
From events 1 and 3 we find ux' = Δx'/Δt' = -γβct/γt = -βc = -v.
From events 1 and 3 we find uy' = Δy'/Δt' = ut/γt = u/γ.
(b) In S' at t' = 0.5γβ/c the left edge of the stick is at x' = -0.5γ, y' =
0.
At t' = -0.5γβ/c the right edge of the stick is at x' = 0.5γ, y' = 0.
The stick travels with velocity u' = -vi + (u/γ)j.
It takes the stick ∆t' = γβ/c to cross the x-axis. In this time interval it
travels a distance u'∆t' = uβ/c meter in the y'-direction.
(c) At t' = 0.5γβ/c the right edge of the stick is at x' = 0.5γ - vγβ/c, y'
= uβ/c.
For the length of the stick in x' we have
Lx' = xR' - xL'
= γ(1 - β2) m = 1/γ m, Ly' = yR'
- yL'
= uβ/c m.
(Note: we have to measure the position of both ends at the same time in S'.)
The length of the stick in S' is (x'2 + y'2)½ =
(1 - β2 + (uβ/c)2)½ m.
The angle θ that the stick makes with the x'-axis is found from tanθ' = Ly'/Lx'
= γuβ/c.
A relativistic particle is launched at the origin (0,0) with initial momentum
p(0) = (px(0), py(0)), with px(0)
> 0 and py(0) > 0, and is subject to a
constant force pointing in the negative y direction.
(a) Solve the equations of motion for x(t) and y(t).
(b) Determine the time T at which the particle reaches the x-axis
again (i.e. y(T) = 0).
(c) Find the trajectory of the particle, i.e. y = y(x).
NOTE: Give all answers for the laboratory frame.
Solution:
Consider an electron propagating in empty space with speed v, and hence with energy Ee = γmc2. Show that the electron cannot emit a photon.
Solution: