π^+ mesons. A burst of 10^4 π^+ mesons travels in a circular path of radius 20 m at a speed β=0

step 1 Time dilation affects particle decay. Here are the particles that are decaying are Pi -Masons, a

Π^+ mesons. A burst of 10^4 π^+ mesons travels in a circular...

$\pi^{+}$ mesons. A burst of $10^{4} \pi^{+}$ mesons travels in a circular path of radius $20 \mathrm{~m}$ at a speed $\beta=0.99 c .$ The proper mean life of the $\pi^{+}$ meson is $2.5 \times 10^{-8} \mathrm{~s}$. (a) How many survive when the burst returns to the point of origin? (b) How many mesons would be left in a burst that had remained at rest at the origin for this same period of time?

$\pi^{+}$ mesons. A burst of $10^{4} \pi^{+}$ mesons travels in a circular path of radius $20 \mathrm{~m}$ at a speed $\beta=0.99 c .$ The proper mean life of the $\pi^{+}$ meson is $2.5 \times 10^{-8} \mathrm{~s}$.
(a) How many survive when the burst returns to the point of origin?
(b) How many mesons would be left in a burst that had remained at rest at the origin for this same period of time?

Key Concepts

Exponential Decay

Exponential decay is the process by which unstable particles diminish in number over time, characterized by the formula N = N? exp(?t/?), where ? is the characteristic lifetime. This concept is crucial for understanding how many particles remain after a certain period, regardless of the motion of the system.

Time Dilation in Special Relativity

Time dilation is a fundamental result of special relativity showing that moving clocks run slower compared to stationary ones. For particles moving at relativistic speeds, their lifetime as measured in the laboratory frame is extended relative to their proper lifetime, which is measured in their rest frame.

Proper Lifetime vs. Dilated Lifetime

The proper lifetime refers to the lifetime of a particle measured in its rest frame, while the dilated lifetime is the lifetime measured in a different frame where the particle is moving. This concept is essential when comparing decay rates for particles in motion with those at rest.

Circular Motion Dynamics

Circular motion dynamics involve calculating the time taken for a particle to complete a circular path, which is determined by the circumference of the circle and the particle's speed. This calculation is crucial for figuring out the time interval over which the particle decay process occurs in the lab frame.

Transcript

00:01 Time dilation affects particle decay.

00:04 Here are the particles that are decaying are pi -masons, and they are traveling at 0 .92c with respect to the lab.

00:16 So recognize that the proper time is in the frame of the pi -mazons, that is, whatever governs their decay is with them and not in the lab.

00:28 So the observed lifetime in the lab is going to be time dilated by factor gamma, and we'll see what consequence that has in our observations.

00:49 But let's first of all figure out what gamma is.

00:53 So we'll do the usual calculation, and that turns out to be 2 .55.

01:03 So what that means is in the lab, the mean lifetime of these pie mesons is going to be two and a half, a little bit over two and a half times longer than the 26 nanoseconds that is true in the pi maison frame.

01:30 And so that is 66 .3 nanoseconds.

01:34 So what we would expect is that the number of pymeisons that survive after a certain amount of time in the lab, so this is observed in the lab, is going to be exponentially decaying, and we are going to let the piemaeons travel a 50 -meter distance in the lab, in lab frame s.

02:14 So that means that that is a proper length.

02:16 Now, of course, to the pi mesons, that 50 meters is going to be foreshortened according to lorentz contraction.

02:28 But using that distance and the velocity of these as 0 .9c, we can determine the amount of time that it would take for them to travel the 50 meters, and we can use that up in our time decay and see how many we would expect to survive after the 50 meter long trail.

02:59 So we will divide that by 0 .92 times i like to use lots of sick figs in my speed of light, so that time is going to be...

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