A spaceship flies past an experimenter who measures its...

Name: Problem 51, Chapter 27: College Physics: A Strategic Approach
Uploaded: 2020-11-29T19:11:30-08:00
Duration: 2 min 57 s
Channel: Guilherme Barros
Description: A spaceship flies past an experimenter who measures its length to be one-half the length he had measured when the spaceship was at rest. An astronaut aboard the spaceship notes that his clock ticks at 1-second intervals. What is the time between ticks as measured by the experimenter?

A spaceship flies past an experimenter who measures its length to be one-half the length he had measured when the spaceship was at rest. An astronaut aboard the spaceship notes that his clock ticks at 1-second intervals. What is the time between ticks as measured by the experimenter?

Key Concepts

Length Contraction

Length contraction is the phenomenon predicted by special relativity in which the length of an object moving relative to an observer is measured to be shorter than its proper length (the length measured in the object's rest frame). This effect, like time dilation, becomes significant at speeds close to that of light and is intrinsically related to the Lorentz transformations.

Lorentz Factor

The Lorentz factor, typically denoted by gamma (?), quantifies the effects of time dilation and length contraction in special relativity. It is defined as 1/?(1 - v²/c²) and provides the scaling between proper time and the time observed in a moving frame, as well as between rest length and contracted length.

Special Relativity

Special relativity is a theory developed by Einstein that describes the physics of objects moving at speeds close to the speed of light. It introduces the idea that measurements of time and space are relative to the observer's inertial frame and leads to phenomena that seem counterintuitive when compared to classical mechanics.

Proper Time

Proper time is the time interval between two events as measured by an observer for whom the events occur at the same location. It is the shortest time interval between those events and serves as a reference for understanding time dilation in different inertial frames.

Time Dilation

Time dilation is the phenomenon where the elapsed time between events is longer in a moving reference frame compared to the proper time measured in the rest frame. This effect increases as the relative speed approaches the speed of light, and it illustrates one of the key counterintuitive results of special relativity.

Transcript

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

00:04 So first, about length contraction, consider that we have an object that has a proper length l.

00:09 This is the length as measured in the reference frame of the object, that is in the reference frame where the object is at rest.

00:19 And consider that this object is moving with the speed v relative to an observer at rest with reference frame s.

00:26 This observer will measure for the length of the object, the proper length l divided by gamma, where gamma is 1 divided by the square root of 1 minus g squared over c squared.

00:41 Secondly, let's talk about time dilation.

00:44 So consider that we have an s reference frame, where two events happen in the same place, but at different times.

00:54 And the time measure between these two events is called tau.

00:57 This is the proper time.

01:00 And it's called the proper time because it's the time between two events that happen in the same place.

01:10 Now consider a second reference frame as prime according to which the first reference frame is moving with a speed v.

01:18 Now notice that the two events don't happen in the same place anymore.

01:21 And the time between the two events, according to spectral activity, is equal to the proper time times gamma, where gamma follows the same expression as before...

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