A rocket ship flies past the earth at 91.0% of the speed of li...

Question

A rocket ship flies past the earth at 91.0% of the speed of light. Inside, an astronaut who is undergoing a physical examination is having his height measured while he is lying down parallel to the direction in which the ship is moving. (a) If his height is measured to be 2.00 m by his doctor inside the ship, what height would a person watching this from the earth measure? (b) If the earth-based person had measured 2.00 m, what would the doctor in the spaceship have measured for the astronaut’s height? Is this a reasonable height?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Technological innovations in an alien's planet have facilitated the evolution of spacecrafts that move at nearly the speed of light. A 1.00 m long meter ruler as measured by a scientist in a spacecraft is placed parallel to the direction of the spacecraft's motion. The spacecraft flies past the aliens planet at 82.5% of the speed of light. A find the ruler's length as measured by a scientist on the aliens planet. A scientist on the aliens planet measured the length of a different ruler in the spacecraft to b 1. m. B what is the ruler's length measured by a scientist in the spacecraft? OK. So our end goal is to find two separate answers. Our first answer is to find the ruler's length as measured by a scientist on the aliens planet. And then our second answer we're trying to find is what is the ruler's length measured by a scientist in the spacecraft. OK. So we're given some multiple choice answers. We're given answers for part A and for part B and they're all in the same units of meters. So let's read them off to see what our final answer pair might be. A is 0.565 and 1.77 B is 1.30 and 0.565 C is 0.565 and 1.30 D is 1.30 and 0.771 E is 1.77 and 0.565. And F is 0.771 and 1.30. OK. So first off, we have to note, we need to make the following note that we're told that the ruler is positioned parallel to the direction of the spacecraft's velocity. Therefore, the ruler's length will be Lourens contracted. Also, we need to note that the proper length of the ruler is I zero. So let's write that down really quick. So I zero equals proper length. So, so the proper length of the ruler is I zero and it will be measured by the scientists in the spacecraft. So now we need to recall and use the equation for relativity to write. And let's call it equation one that I is equal to I zero, I subscript zero multiplied by the square root of one minus U squared divided by C squared. OK. So let's make a couple of notes really quick again. So note that I zero is equal to 1.00 meters and that you is equal to 82.5% C which we need to write the percentage as a decimal. So all we have to do is just take 82.5 divided by 100. So it's the same as saying 0. C Fantastic. So now we can plug in all of our known variables to solve for I. So let's do that. So I is equal to 1.00 m multiplied by the square root of one minus U which was 0.825 C squared divided by C squared. Note that the C squares cancel out leaving us with just the units of meters, which is good means we know we remember that all of our answers for the multiple choice are in meters. So that's a good sign we're on the right track. So when we plug that into a calculator, we should get 0. meters. And this is the length of the ruler measured by a scientist on the aliens planet. OK. So let's solve for part B you can make sure. So we don't get confused. So that's part A. So let's start solving for part B. We need to use equation one and rearrange it to solve for I zero. In this case So I zero, when you use a little bit of algebra to rearrange equation one, we get that I zero is equal to I divided by the square root of one minus U squared divided by C squared. So now like above, we need to plug in all of our known variables to solve for I zero. And OK. So and above, just in case we're confused because we're like, wait a minute, I zero is supposed to be 1.00 m. Well, that was all nice and dandy for part A because it's the length of the ruler measured by the scientists on the aliens planet. But now for part B, we're trying to find the length of the ruler measured by a scientist in the spacecraft. So the frame of reference is different. Thus I zero is a different value. So that's why we had to rearrange insult. So let's plug in all of our known variables. We know that I in this case is also 1.00 m as given to us in the problem. And we need to take that and divide it by one minus U which was 0.825 C squared divided by C squared like above C squared cancels out. So when we plug into a calculator, we get 1.77 m. So that means our final answer or B is 1.77 m and this is the length of the ruler measured by a scientist in the spacecraft. Fantastic. We did it. So looking at our multiple choice answers, that means the correct answer has to be the letter A A is 0.565 m and B is 1.77 m. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Special Relativity

Length Contraction

Lorentz Transformation

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