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A spacecraft with a proper length of 300 $\mathrm{m}$ takes 0.750$\mu \mathrm{s}$ to pass an Earth-based observer. Determine the speed of the spacecraft as measured by the Earth observer.

$v=0.8 c$

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Cornell University

Rutgers, The State University of New Jersey

Numerade Educator

Hope College

In this exercise, we have a spacecraft that has a proper link of 300 m. And this brother Erling, I'm calling V. That takes a time of zero 0.75 microseconds. In order to pass an observer that is standing on the earth. Our goes to find what is the speed of this aircraft. According to the observer. On your so, first of all, we need to review some things. The first one is that the proper length is the length. According to the Observer, that is inside the spacecraft, meaning that the distance de that length that the observer on the Earth will measure for the spacecraft is different from the proper length. And we have a formula for that. We know that the distance D that will be observed by the observer on the earth. It's equal to the proper distance de times uh, divided by Goma. We're gonna is one over the square root of one Mina's View square oversee square. OK, notice that the the distance that we will be observed by the observer on the Earth is necessary necessarily smaller, then the proper length off the spacecraft a meaning that the proper ling is always maximum. Also, the speed of the spacecraft, according to the Observer, own the earth will be the distance that the observer on the Earth measures divided by the time that it matters. So basically, all we have to do in order to calculate V is to insert this warm without here into the formula for velocity and van. We care. Find what? The ISS. So let's do it. We have a V sequel to D divided by T, but this lower case D here is able to the upper case T that is the proper distance. Times one minus B squared over C squared. Then we have two divided by t. So I'm gonna multiply both sides by teeth. You have V t is equal to d times the square root of one miners view squared overseas square and we can square both sides together. The square T square is equal to the square times one miners v squared over C squared and I'm gonna try to isolate Visa. Get that B squared times T square was d squared, divided by C Square. It's equal to d swear. So I'm gonna again keep trying to as late visa have B squared is equal to D squared, divided by t squared was D squared overseas square. And if we take the square root, we get that the is equal to D, divided by the square root of T square less Do you oversee squid and I can simply insert the numbers so be that is a proper length is 300 m t is 0.75 microseconds. That's 7.5 times 10 to the minus seven seconds. So there we have 300 m, divided by the speed of light. That's three times 10 to the eighth meters per second. So V is equal to 2.4 times 10 to the 8 m per second, which is the same as your 0.8 times this beautiful light. See? So this is the speed of the spacecraft as it passes through, uh, the

Universidade de Sao Paulo

Gravitation