Refer to Example 26.2. One million muons are moving toward...

Refer to Example 26.2. One million muons are moving toward the ground at speed $0.9950 c$ from an altitude of $4500 \mathrm{~m}$. In the frame of reference of an observer on the ground, what are (a) the distance traveled by the muons; (b) the time of flight of the muons; (c) the time interval during which half of the muons decay; and (d) the number of muons that survive to reach sea level. [Hint: The answers to (a) to (c) are not the same as the corresponding quantities in the muons' reference frame. Is the answer to (d) the same?]

Key Concepts

Lorentz Factor

The Lorentz factor, denoted by ?, quantifies the degree of time dilation and length contraction experienced by an object moving at relativistic speeds. It is calculated based on the ratio of the object’s speed to the speed of light and serves as a scaling factor for transforming time and distance measurements between different inertial frames.

Time Dilation

Time dilation is the relativistic effect whereby a moving clock is observed to tick slower compared to a clock at rest. This phenomenon is essential for analyzing how measured time intervals, such as the decay lifetime of muons, differ when observed in different reference frames.

Length Contraction

Length contraction is the reduction in the measured length of an object in the direction of motion as observed from a stationary frame relative to the moving object. This effect explains why distances measured in the muons' frame differ from those measured in the ground frame.

Exponential Decay

Exponential decay describes the process by which the number of unstable particles, like muons, decreases over time according to a characteristic decay constant or half-life. This concept is crucial for determining the fraction of muons that survive over a given time interval as they travel.

Invariance of Event Counts

While time and distance measurements are relative in different reference frames, the total number of events, such as the number of surviving muons reaching sea level, is invariant. This principle ensures that certain outcomes, like particle counts after decay, are consistent across all inertial frames.

Transcript

00:01 For question a, the distance is just simply equal to the altitude of the mountain, which is 4 ,500 meters.

00:06 Okay? because this distance is measured by the observer from the earth, so it's still equal to 4 ,500 meter.

00:13 And for question b, it was asking us the time of flight.

00:17 Well, we know the distance is equal to 4 ,500 meter.

00:25 And we also know the speed of the mules.

00:29 Is given as 0 .9950 c.

00:39 Okay? and then we'll see the speed of lights.

00:42 And therefore we can determine the time, which is equal to the distance divided by the speed, which is equal to 4 ,500 meter divided by the speed, which is 0 .9950c.

01:01 And c is the speed of light, which is 3 .0 times 10 to a bit.

01:06 Power of 8 meter per second.

01:10 Since 3 and 3 .0 is the same, so it's right there 3.

01:14 Okay, it doesn't really matter.

01:20 So, and this will give us the time is about 1 .5 times central power of negative 5 seconds.

01:30 If we convert in microseconds, it's 15 microseconds.

01:34 Okay? so this is the time of the flight.

01:39 And then for question c, what's the time of the flight from bion's perspective? in another word in a miwan's red frame.

01:50 Well, we know we can come up with such equation, which is dollar t is equal to gamma, i'm sorry, we should be delta t here.

02:02 Della t is equal to gamma times delta t.

02:08 So therefore, since we're trying to determine delta t zero, so we have delta t zero, is equal to delta t divided by gamma.

02:24 We know gamma is the variance factor, which is 1 over square root, 1 minus v squared over c squared.

02:33 Okay, and we know delta t is given from the last question, which is 15 microseconds, okay? so 15 microsecond divided by gamma, which is 1 over square.

02:57 Square root 1 minus v square...

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