A star is 23.5 light-years from Earth. How long would it take ...

Question

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers: What is the distance traveled according to observers on the spacecraft?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A planet is located 50 light years away from a space station. A probe is sent towards the planet at a speed of 0.80 multiplied by C the observers on the probe measure the time taken to reach the space station to be 37.5 years determine as measured by the observers on the probe I the distance to the planet. And I I the velocity of the probe, please write your final answers in the units of the speed of light. Awesome. So it appears for this particular problem, our final answer or answers that we're ultimately trying to solve. For our end goal is we're trying to figure out the, we're trying to determine as measured by the observers on the probe. So we're focusing on the perspective or the frame of reference of the observers that are located on the probe. And we're trying to figure out for I the distance to the planet. And for I, I, we're trying to figure out the full velocity of the pro and for both I and I, I, so both of our answers, we're focusing on the frame of reference or rather the perspective of the observers that are on the probe, which for relativistic problems like that, it's very important to figure out where the point of reference is awesome. So with that in mind now that we know what we're ultimately solving for, let's read off our multiple choice answers to see what our final answer pair might be. Noting that for all of our answers, for part I are in units of light years and all of our answers for part I, I are some value multiplied by C which like the problem itself. And all of our multiple choice answers C denotes the speed of light. So it's some value multiplied by the speed of light. So A is 30 0.62 B is 30 0.80. C is 63 and 0.62 and D is 63 and 0.80. OK. So first off, let us write down all of our known variables. So we know the value of the distance from the space station to the planet and the distance from the space station to the planet will call this value D is equal to 50 light years. Fantastic. We also know the speed of the probe which we'll call V and the speed of the probe is equal to 0.80 multiplied by C. And we also know the time taken to reach the planet with respect to the space probe observers, we'll call this value T zero and T zero is equal to 37.5 years. Fantastic. So moving right along here. So our first step is we need to calculate the Lorenz factor. So we need to recall and use the Lorenz factor equation which states that gamma, which is the Lorenz factor is equal to one divided by the square root of one minus V squared divided by C squared. Fantastic. So when we plug in our known variables, we will take one divided by the square root of one minus our speed which is zero point 80 C squared divided by C squared. Now, as you can see, our speed of light constant will cancel out. So when we plug that into our calculator, we will find that it's equal to 1.667 when we round to three decimal places. So now we could start to officially solve for part I now. So focusing on part I, so we need to recall and use the length contraction in order to help us find the distance as measured by the observers on the probe. So as we should recall, our equation should state that do which is the final answer we're ultimately trying to solve for is equal to D divided by gamma. So when we plug in our known variables, we will take 50 light years divided by our gamma value, which is 1.667. And when we plug that into our calculator, and we round to the nearest pole number or to Yeah. So the nearest phone number we will get 30 light years, which is our final answer. Hooray, we did it. And just in case since we're gonna need this value, let's write our un rounded value two views that help us get a more accurate answer when we use this answer to help us solve. For part I I, so as we should note when we round to three deel places before we round, we will get 29.994 light years initially as our final answer. OK. So now we solve for part I. So let's make a little note here. So our answer for part I is 30 light years. Now we could start solving for part I I. So now we need to recall and use the contracted length which is D zero, which is the value we just found together. And the time value in order to help us figure out the velocity of the probe is I I we're trying to figure out the velocity of the probe. That's our second answer we're ultimately trying to solve for. So the velocity V zero of the probe can be calculated by the observers on it are on the probe itself. And as we should recall, we should use the following equation which considers the perspective of the observers on the probe, which states that V zero, the speed is equal to D zero divided by T zero. Which when we plug in our known variables, we know that D zero is equal to 29.994. Since we're using the unrolled value to get a more accurate answer. So 29.994 light years divided by 37 point five years, which will just, you will abbreviate years to Yrs. So let me plug that into our calculator. We will get 0.7998 multiplied by C which will mean that V zero is approximately equal to when we round to two decimal places, 0.80 multiplied by C and that is our final answer. Hooray, we did it. And once again, it's important to note that C is our speed of light constant. OK. So with that in mind, the final answer has to be multiple choice answer letter B which states that I is 30 light years and I, I is 0.80 multiplied by C. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video.

A star is 23.5 light-years from Earth. How long would it take a spacecraft traveling 0.950c to reach that star as measured by observers: What is the distance traveled according to observers on the spacecraft?

Key Concepts

Relativity of Time and Distance

Lorentz Transformation

Speed Calculation in Relativity

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