A cosmic ray travels 60 km through the earth's atmosphere in...

Key Concepts

Proper Time

Proper time is the time interval measured by an observer moving along with an object, essentially recording the time experienced in the object's own rest frame. It is the smallest possible time interval between two events and is unaffected by the time dilation observed in other frames.

Lorentz Factor

The Lorentz factor, often denoted by ? (gamma), is a critical component in special relativity that quantifies the relationship between time dilation, length contraction, and the relative velocity of two inertial frames. It is defined as ? = 1/?(1 - v²/c²), where v is the relative velocity and c is the speed of light, and is essential for calculating how much time 'slows down' for moving observers.

Special Relativity

Special relativity is the framework that describes how observers in different inertial frames measure time and space differently. It establishes that the speed of light is constant in all inertial frames, leading to effects such as time dilation and length contraction when objects move at significant fractions of the speed of light.

Time Dilation

Time dilation is a consequence of special relativity which indicates that a clock moving relative to an observer will be measured to tick slower than a clock at rest with respect to that observer. This effect becomes significant as the relative velocity approaches the speed of light.

Transcript

00:01 In this problem, we're going to talk about time dilation.

00:02 So consider that we have a reference frame.

00:05 I'm going to call it the s reference frame.

00:09 And according to this reference frame, two events happen in the same place.

00:13 And the time between these two events is called tau.

00:17 Tau, i'm sorry, this is the proper time between the two events.

00:21 And it's special because it's the smallest possible time measured between these two events.

00:28 And you can recognize it because it's.

00:30 The time measure between the two events in the reference frame where the two events happen in the same place.

00:37 Now suppose that we have a second reference frame, i'm going to call it s prime, and in this reference frame the s reference frame is moving with a certain speed v.

00:50 Now notice that the two events don't happen in the same place anymore, they happen separately, and the time t between the two events is measured by the s prime reference ring is equal to tau times gamma, where gamma is 1 over the square root of 1 minus v squared over c squared.

01:11 Okay, what we have in our problem is a cosmic ray that travels a distance of 60 kilometers in a time of 400 microseconds according to an earth observer.

01:27 And our goal is to find how much time passes according to an observer at rest relative to the cosmic ray.

01:36 Well, first, notice that the speed of the cosmic ray relative to the earth is equal to d divided by t, so that's 60 kilometers, or 60 times 10 to the 3 meters, divided by 400 microseconds, or 400 times 10 to the minus 6 seconds, and this is equal to, v is equal to, 1 .5 times 10 to 8 meters per second...

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