Subatomic particles called pions are created when protons, accelerated to speeds
ID: 3278085 • Letter: S
Question
Subatomic particles called pions are created when protons, accelerated to speeds near the speed of light in a particle accelerator, smash into the nucleus of the target atom. Charged pions are unstable particles that decay into muons with a half-life of 1.8 x 10-8s. Pions have been investigated for use in cancer treatment because they pass through tissue with minimal damage until they decay. As they decay the release significant energy. The speed of the pions can be adjusted such that the most likely place for the decay is in the tumor.
Suppose the pions are are created in an accelerator then directed into a medical bay 30 m away. They travel at a speed of 0.99995c. Without time dilation, half the particles would have decayed after traveling only 5.4 m, not far enough to make it to the medical bay. Time dilation allows them to survive until they get to the tumor.
What is the half-life of the pion in the reference frame of the patient?
According to the pion, what is the distance the pion travels from the accelerator to the medical bay?
The proton collision also creates a gamma ray photon that travels in the same direction as the pion. The photon will get to the medical bay first because it is traveling faster. What is the speed of the photon in the the pion's reference frame?
If the pion slows down to 0.99990c, what percentage of kinetic energy is lost?
Explanation / Answer
1. half life of pion, to = 1.8*10^-8 s
speed of pion, v = 0.99995c
so, half life as seen by the patient, t = to/sqroot(1 - v^2/c^2) = 1.8*10^-8/sqroot(1 - 0.99995^2) = 1.8*10^-6 s
2. distance to the bay in inertial referencef frame, lo = 30 m
length seen by pion, l = lo*sqroot(1 - v^2/c^2) = 30sqroot(1 - 0.99995^2) = 0.29999 m
3. speed of photon in pion's reference frame = c [ as photons always travel at speed c, no matter what reference frame it is ( from special theory of relativity)]
4. Relativistic kinetic energy is given by
E = mc^2 - moc^2 [ where mo is the rest mass ]
but m = mo/sqroot(1 - v^2/c^2)
so, KE = mo*c^2/sqroot(1 - v^2/c^2) - moc^2 = moc^2 [ 1/sqroot(1 - v^2/c^2) -1]
initial velocity, v = 0.99995c
final velocuty, u = 0.0000c
so, percentage KE lost = { [ 1/sqroot(1 - v^2/c^2) -1] - [ 1/sqroot(1 - u^2/c^2) -1]}/ [ 1/sqroot(1 - v^2/c^2) -1] ={ 1 - sqroot(1 - 0.99995^2)/sqroot(1 - 0.9999^2)} = 0.292
E perentage lost = 29.288 percent