The distance to a particular star, as measured in the earth's frame of reference, is 7.11 light-

step 1 This is chapter 37 problem number 52. We're given the distance from Earth to a star, 7 .11 light

The distance to a particular star, as measured in the...

Key Concepts

Special Relativity

Special relativity is the framework that describes how space and time are interwoven and how they are perceived differently by observers in different inertial frames. It asserts that the laws of physics are the same in all inertial frames and introduces the constant speed of light as a fundamental limit for the propagation of information and matter.

Time Dilation

Time dilation refers to the phenomenon where clocks moving relative to an observer are measured to tick slower compared to clocks at rest in the observer's frame. This effect becomes significantly noticeable at speeds close to the speed of light and is a direct consequence of the invariance of the speed of light in all inertial frames.

Length Contraction

Length contraction is the observed shortening of the measured length of an object moving relative to an observer, specifically along the direction of its motion. This relativistic effect says that lengths measured in the moving frame will be contracted compared to the object's proper length, which is measured in its own rest frame.

Lorentz Factor

The Lorentz factor, typically denoted as ? (gamma), quantifies the degree of time dilation and length contraction experienced by an object moving at a significant fraction of the speed of light. It is defined by the formula ? = 1/?(1 - (v²/c²)), where v is the relative velocity and c is the speed of light, and plays a crucial role in relating physical measurements between different inertial frames.

Transcript

00:01 This is chapter 37 problem number 52.

00:03 We're given the distance from earth to a star, 7 .11 light years.

00:10 And the time for an observer on the spaceship is given to us.

00:15 Let me write it down 3 .35 years.

00:19 The distance to a star, let's write it down distance x, let's call it x from the earth.

00:27 Right, it's 7 .11 light years.

00:32 So we need to just figure out what's the proper time and what's the proper length or distance first, right? so in this case, for an observer on the spaceship, if this is the time measured, since the observer happens to be at the same position at the start and the finish of this travel, then this time has to be the proper time.

00:57 So the t is going to be our proper time.

01:05 And if the distance from the earth to the star is given to us as 7 .1 light years, and this is the actually proper distance.

01:23 Okay? so let's try to figure out what the time would be for an observer on earth.

01:32 If our proper time is the time of a passenger in a spaceship.

01:40 So the t prime, we're going to call it the time on earth.

01:44 Time on earth.

01:47 With respect to the proper time, then, is going to be gamma times t, the proper time.

01:53 As you know, gamma here is 1 over square to 1 minus a v squared over c squared.

01:57 And we're not giving this information, v, being the relative speed of the, of a ship, right? we need to figure this out first.

02:06 And then our answer to part a is here embedded in this equation.

02:11 And then in order to get to v, we need to know time as well as the distance.

02:17 Now, if the distance from earth to the star is given to us, which is the proper distance, and we're trying to figure out x prime then is going to be x over gamma again.

02:34 Our answer to park b is admitted here, but first we need to figure out what v is, since both of these equations involve gamma.

02:43 If you don't know what v is, we don't know what gamma is, and we can't answer the question.

02:47 So now, how are we going to calculate the relative speed of the ship? relative speed of the ship is going to be equal to the proper distance, right? which is x from the earth to that star, divided by the time.

03:04 That passes for the people on earth, right? so it's going to be x over t prime.

03:11 This is the velocity that we're that we're looking for.

03:15 And now if that's what we're looking for and we know what x is from this equation, then from here, well x is actually given to us, so i'm not going to go back to x prime.

03:26 X is given to us t prime however is gamma times t.

03:31 So i can rewrite this equation as x over gamma times t here in this equation i know what x is and i know what t is and i know i have a gamma factor here which involves v and also i have v here now let's plug in the numbers and try and calculate what the what the speed is right so v is going to be equal to x is 7 .11 light years so we're going to do 7 .11, right, times the speed of light then, divided by the gamma factor times 3 .35 years.

04:14 So if we rearrange things a bit, then what we have is, well, actually, let's divide both sides by c, okay? so let's get it over the c in the right hand.

04:28 So v over c from here, then it's going to be equal to 7 .11 divided by 3 .35 and 1 over gamma.

04:39 Now let's figure out what this is.

04:41 If you use your calculator, you're going to see that it looks something like, actually we could put the equal sign here, equals to 2 .12.

04:57 Times 1 over gamma.

04:59 Still, the left -hand side is v over c...

Please give Ace some feedback