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Problem 17 Medium Difficulty

A spacecraft with a proper length of 300 m passes by an observer on the Earth. According to this observer, it takes 0.750$\mu$ s for the spacecraft to pass a fixed point. Determine the speed of the spacecraft as measured by the Earth-based observer.

Answer

0.800$c$

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Video Transcript

well. The contracted land, measured by an observer on the earth is given by L is equal to LP, divided by gamma. Elope is proper Lent In common is the relativistic factor. Well, let Ristic factor got my is equal to one divided by route one minus re squared, divided by C square And in terms off Lenk we have l is equal to has be into square root off one minus every square divided by C square By substituting gum line on this equation we get Al is equal to lp Times square root off one minus B squared, divided by C square and taking screwed on both sides. L Square is equal to L P Square into one minus Reese created by the by C Square. And let's say this is the question number one Since l is equal to three times Delta t we are given Delta T Well, Delta T is equal to 0.750 Multiply bite into the power minus six seconds. Right and proper length MP equals 300 meter. And now let's substitute these values iniquity Number one, Then we get we square. We re square into 0.75 c Don't multiply by 10 to the power my six hold square. Right, um, is equal to 300 square treating 100 squared into one minus B square, divided by C square. Well sees three. Multiply Bite into the power eight old square. All right, now, further, we square Greece were into 0.75 Siegel 0.750 multiplied by 10 to the power minus six. Hold square is equal to nine. Multiplied by 10 to the power four into one minus re square divided by nine months. Supply by 10 to the power 16. Right. And further simply flying, we get 1.56 to 5. Noted, like by 10 to the power minus 12. Re square is equal to 5.7 six month supply by 10 to the power 16. And so we is equal to we have, uh, here. Uh, Lee is equal. Do, uh, 20 Fuller. Most of lie by 10 to the power seven meter per second. Now dividing by sea, let's divide this by sea. So a year redivided by sea. And then we also divide this by three. Multiply by 10 to the power eat meter per second. Then we divided by C, is equal to it multiplied by 10 to the power minus one. Right, So we divided by C, is equal to we have seen a 0.8, So we is equal to 0.8 times the speed of light.

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