In an attempt to reduce the extraordinarily long travel times ...

Question

In an attempt to reduce the extraordinarily long travel times for voyaging to distant stars, some people have suggested traveling at close to the speed of light. Suppose you wish to visit the red giant star Betelgeuse, which is 430 ly away, and that you want your 20,000 kg rocket to move so fast that you age only 20 years during the round trip. How fast, as a fraction of c, must the rocket travel relative to earth?

Accepted Answer

Hey, everyone in this problem, we're told that a spacecraft of mass 18,500 kg travels from earth to explore a planet located 650 light years away. The astronauts on the spacecraft aged 25 years during the journey from earth to the planet. And we're asked to determine the speed of the spacecraft in terms of C relative to the earth. Option A is 0.89752 C option B 0.99015 C option C 0.99926 C and option D 0.99968 C. All right. So let's start by just defining our reference rate and we have a problem here where we're told some information about the spacecraft. OK. And what the astronauts on that spacecraft observe and we're asked for the speed relative to the earth. OK. So we have two kinds of reference frames here. So we're gonna say that the earth reference frame is going to be frame S and that the spacecraft reference frame is going to be frame S prime. OK. So we've just defined our two reference frames and we're gonna say that frame s prime moves at some speed which we're gonna call V relative to us. OK. So the speed that our spacecraft frame is moving relative to the earth frame. Well, that's gonna be the same as the speed of the spacecraft relative to the earth. OK. So that speed V is actually what we're looking for in this problem. That's what we wanna solve for. Now, we're getting some information about time here and the time that's elapsed on the spacecraft. OK. So let's go ahead and think about our time dilation. So we know that we have delta T equal to delta T prime divided by the square root of one minus beta squared. OK. Where delta T is a time elapsed in our earth frame, delta T prime is a time elapsed on the spacecraft frame. Now, looking at this as it is, it doesn't really, it's not clear how this relates to our speed V. But recall that we can write beta as V divided by C. And so we can write delta T as delta T prime divided by the square root of one minus V squared divided by C squared. And so now that value of V that we're looking for is in our equation. Now, what about these other values? Well see is the speed of light that's a constant. We know that value delta T prime were given. OK. We're told that the astronauts on the spacecraft aged 25 years. So that tells us that delta T prime. And let's write this in another color. So it's clear delta T prime is going to be equal to 25 years. OK. All right. So that's everything on the right hand side that we know on the left hand side, we have delta T. Now delta T is going to be equal to one or recall that our speed V is equal to distance divided by time. And so we can write that delta T is equal to the distance D divided by the speed V. OK. Remember this is our earth frame. Delta T that we're looking for. So that's gonna be the speed relative to the earth. OK. So that's the speed V that we're looking for. When we're doing these problems, we always have to make sure that the distance, the time and the speed we're dealing with are in the same frame of reference. OK? All right. So we know the distance we're told that this is 650 light years away from earth. OK? So delta T is gonna be 650 light years divided by V. OK? So we don't have an exact value for delta T right now, but we can write it in terms of this speed V. And if we do that, that speed V becomes the only thing in our equation that's unknown and we can solve work. All right. So let's do that, let's substitute everything we just talked about into this equation. OK. So delta T that's gonna be 650 light years divided by V. This is gonna be equal to delta T prime, which is 25 years divided by the square root of one minus V squared divided by C squared. Now, all of our values are in terms of light years and years. So when we write C, we're gonna write it as one light year per year instead of writing it in terms of meters per second, we want all of our units for time and distance to be consistent so that it's easier to simplify our equation. All right. So this looks a little messy, but again, it's just an equation where we have one unknown V that we wanna solve. OK? Now, don't get confused with these. Why's OK? Sometimes we can confuse them and think that it's a variable. But why is just the unit here and there a unit for time here? All right. So first things first, let's get rid of this square root. We're gonna square both sides. And when we do that, we get 422,000 500 light years squared, divided by V squared is equal to 625 years squared, divided by one minus V squared, divided by one light year per year squared. All right, we're gonna multiply both sides by V squared. We're gonna multiply both sides by this one minus V squared divided by C squared term. And when we do that, we won't have these denominators, messy denominators to deal with. So we get 422,500 light years squared, multiplied by one minus B squared, divided by one light year squared per year squared is equal to 625 years squared multiplied by V squared. All right, we're getting closer, but we still have a V term on both sides. So let's expand on the left hand side. Simplify as much as we can. We wanna get all of these V terms to one side. OK. So first step, let's divide by this 625. OK. When we do that, we get 676 light year squad multiplied by one minus V squared, divided by one light year squared per year squared. Ok? And that's gonna be equal to B squared. Now, we're also dividing by this unit of year squared K. On the right hand side, we don't have this unit left and so on the left hand side, our unit needs to be light year squared per year squared. I should be very careful with that when you're dividing that you bring your units over as well. All right. Now, let's expand on the left hand side. OK. The 676 needs to be applied to everything in the brackets. So we get 676 light years squared per year squared minus 676 multiplied by V squared. That's gonna be equal to V squared. Ok. And notice that our units on the 676 in the second term have gone away. We had light year squared per year squared, divided by light year squared per year squared. Ok. So those units divide out. So we're left with 676 no units there. All right. Now, we can move 676 V squared to the right hand side. We're gonna add it. We get 677 V squared is equal to 676 light years squared per year squared. This tells us that V squared is gonna be equal to 676 divided by 677 what year squared per year squared, right? If we take the square root and we simplify, forget that V is going to be 0.99926 light years per year and recall that one light year per year. Well, that's equivalent to C so we can write this as V is equal to 0.99926. All right. So we found our speed and let's remind ourselves what we did that looked really messy. There was a lot of messy simplification that we needed to do. But the key to this problem was figuring out how to write our time dilation in terms of the variable V that we were interested in solving for. And once we did that, it was just a matter of simplifying that expression, comparing what we found to our answer choices, we can see that option C is the correct answer. 0.99926 C is the speed of the spacecraft relative to earth. Thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

Relativity of Time

Speed of Light (c)

Distance and Time Relationship in Space Travel

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