The nearest star to Earth is Proxima Centauri, 4.3 light-years...

Question

The nearest star to Earth is Proxima Centauri, 4.3 light-years away. At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 5.5 years, as measured by travelers on the spacecraft?

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealing with special relativity. This problem says a science fiction writer is creating a story about a journey to Barnard Star which is 6.0 light years away to keep it realistic. The writer decides that the crew will experience 6.0 years of travel time aboard the spacecraft. There are two parts to this problem. For part one, what constant velocity is required for the spacecraft to complete the journey in 6.0 years as measured by the crew? And for part two, according to the observer on earth, how long does the journey take? We give four possible choices as our answers. For choice. A for part one, the velocity is 0.71 C. And for part two, the time is 9.4 years for choice B for part one, the velocity is 0.71 C. And for part two, the time is 8.5 years for choice C. For part one, the velocity is 0.45 C and the time for part two is 7.1 years. And for choice D for part one, the velocity is 0.45 C. And for part two, the time is 3.5 years. So for part one, we need to determine the speed of the spacecraft and the speed of the spacecraft is going to be the same, whether it's measured by the crew who is moving with the spacecraft or an observer on the earth, who is stationary. We're actually going to use these um crew to measure the speed. So according to them, the speed V is gonna be equal to L divided by delta T night, where V is the speed L is the distance from the earth to the star as measured by someone in the crew. And delta T not is the time that that journey takes according to the crew. Now, the problem we're given the time as measured by the crew. However, the distance that we're given is actually measured by an observer on the earth. And so the distance that an observer on the earth measures is actually called the proper distance. And the crew is actually going to be measuring the contracted dis distance or the contracted length for the journey. So recall your formula for length contraction that is L is equal to L knot multiplied by the square root of one minus V squared divided by C squared. Where L is our contracted length, L knot is our proper length. V is the speed of spacecraft and C is the speed of light in vacuum. So since the crew is measuring the contracted length, we can plug this in, in terms of the proper length, which is what we were given in the problem. So plugging this back into our formula for the speed will get V is equal to L knot divided by delta T knot. That fraction is multiplied by square root of one minus V squared divided by C squared. Now, at this point, I need to just solve this equation for V. And the first step to do that is to square both sides to get rid of the square root sine. So I'll get V squared is equal to the quantity of L knot divided by delta T knot in quantity squared. And this could be multiplying the quantity of one minus V squared divided by C squared. So now I need to collect all my V squared terms on one side of the equation, we'll move them all to the left hand side. And to do that, I'll need to first distribute my L not divided by delta T knott quantity squared term. And then after I do that add over my V squared term. And so when I do that on the left hand side, I'll get V squared and I'm gonna factor that out of the remaining terms on the left hand side. And what I'm gonna be left with is gonna be one plus one divided by C squared multiplied by the quantity of L knot divided by delta T knott in quantity squared. And that's gonna be equal to the quantity of L knot divided by delta T knott in quantity squared. So this point to solve for V, I just need to divide over with multiplying V squared and then take the square root. And so when I do that, I'll get V, it's equal to L not divided by the quantity of delta T knot multiplied by the square root of one plus one divided by C squared, multiplying the quantity of L knot divided by delta T knot in quanti squared. So at this point, I can go ahead and plug in values for the L knot and delta T knot. And so have V is equal to my L knot was six light years. And the way we're gonna write that is gonna be six years multiplied by C. So recall that a light year is the distance that light travels in one year. And so that's just gonna be one year multiplied by the speed of light. So here I have six light years. So it's gonna be six years multiplied by the speed of light. And since we want to write our speed as a factor of C, we'll just leave it as C and then that quantity is going to be divided by our time, which was six years according to the crew that could be multiplied by the square root of one plus one divided by C squared, multiplying the quantity or L knot again will have the six years multiplied by C and this could be divided by a delta T knot which is six years. That whole quantity is being squared. Now, in the denominator, there are factors of C squared will cancel out. And then everything else I can plug into my calculator. And then I'll get V is equal to 0.71 C. And here I've just kept two significant figures. That's gonna be my answer for part one. Now, for part two, we need to calculate the time as measured by an observer on the earth. And they're actually going to be measuring the dilated time since the crew is measuring the proper time. So for part two, we need to recall our formula for time dilation. That is delta T is equal to delta TN divided by the square root of one minus V squared divided by C squared. And we actually have values for the T knot and the V now. And so that means we'll have delta T is equal to fortino will have six years. This could be divided by the square root of the quantity of one minus or V squared will have the 0.71 C. And that quantity is gonna be squared and that's gonna be divided by C squared. Again. In the denominator, the C squares will cancel and I can plug the rest of the values into our calculator. And so when I do that, I'll get delta T is equal to 8.5 years. So this could be our answer for part two. So now if I come back up to my answer choices, correct answer choice corresponds to choice B. So just a quick recap of what we've done here. For part one, we used our definition speed, which is distance divided by time. However, the distance that we use was actually going to be measured by the crew, which was the contracted length. And so we had to use our formula four length contraction to write that contracted length in terms of the proper length, which is what we were given. And the problem then we just had to do some algebra to calculate our speed. For part two, we had to use our formula for time dilation to calculate the dilated time, which is what was um measured on the earth. And that was given that we had the proper time which was measured by the crew as well as the speed that we calculated for part one. So we use those values to calculate our dilated time. So I hope that this has been useful and I'll see you in the next video.

The nearest star to Earth is Proxima Centauri, 4.3 light-years away. At what constant velocity must a spacecraft travel from Earth if it is to reach the star in 5.5 years, as measured by travelers on the spacecraft?

Key Concepts

Relativity of Time

Velocity and Distance

Light-Year

Watch next