The radial velocity of M31 toward the Milky Way is about 100 km / sec . If it maintains that spe

step 1 This problem, we're going to talk about length contraction. So consider that we have an object t

The radial velocity of M31 toward the Milky Way is about 100...

The radial velocity of M31 toward the Milky Way is about $100 \mathrm{km} / \mathrm{sec} .$ If it maintains that speed, how long will it take to travel the 700 -kpc distance between the two galaxies?

Key Concepts

Radial Velocity

Radial velocity refers to the component of an object's velocity that is directed along the line of sight from the observer. It is crucial in astronomy for determining how fast an object, such as a galaxy, is moving toward or away from us.

Astronomical Distance Units

Astronomical distances are often measured in units such as parsecs or kiloparsecs, which are based on parallax measurements. These units help express the vast separations between celestial objects effectively, with kiloparsecs (kpc) representing one thousand parsecs.

Unit Conversion

Unit conversion involves changing one set of units to another to perform calculations accurately. In astrophysical problems, converting distance units from kiloparsecs to kilometers, and time units from seconds to years, is essential for consistently relating speed, distance, and time.

Speed-Distance-Time Relationship

The relationship between speed, distance, and time is fundamental in physics and is described by the equation time = distance/velocity. This formulates the basis for estimating the time it would take for an object to travel a given distance at a constant speed.

Order-of-Magnitude Estimation

Order-of-magnitude estimation is a technique used in astrophysics to approximate values quickly and grasp the scale of astronomical phenomena. This is especially useful when assessing the time it takes to traverse enormous cosmic distances, allowing for a rough, yet insightful, calculation.

Transcript

00:01 This problem, we're going to talk about length contraction.

00:02 So consider that we have an object that has a length l in its own reference frame.

00:09 We give it the name the proper length.

00:15 And consider that this object is moving relative to a certain reference frame, i'm going to call it s, with a speed v.

00:27 According to this reference frame, the length of the object is equal to the proper length divided by gamma, where gamma is 1 over the square root of 1 minus v squared over c square.

00:40 Okay.

00:44 So what we have in our problem is that the diameter of the milky way is equal to 100 ,000 light years.

00:56 And we have a spaceship that is traveling across the milky way and measures the length to be equal to only one light year.

01:10 And our goal is to find in question a, what is the speed of the spaceship? so notice that d is the length, the diameter as measured by the spaceship, while capital d is the proper diameter of the milky way.

01:32 It's measured is the diameter as measured in the reference frame where the milky way is at rest.

01:39 So little d is equal to capital d divided by gub.

01:43 This is d times the square root of 1 minus v squared over c squared.

01:50 So we have the d over capital d squared is equal to 1 minus v squared over c squared.

01:59 So v is the square root of 1 minus d over capital d squared times the speed of light c...

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