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Problem 10 Medium Difficulty

A star is 15.0 light - years (ly) from Earth. (a) At what constant speed must a spacecraft travel on its journey to the star so that the Earth–star distance measured by an astronaut on board the spacecraft is 3.00 ly? (b) What is the journey’s travel time in years as measured by a person on Earth and (c) by the astronaut?

Answer

a) $15.01 y$
b) $15.33 y$
c) $3.06 y$

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Video Transcript

well, distance off star from Earth is 15.0 light years. I understand So star distance. Well, Stahler, uh, from Mark is equal to 15.0 light years. Let's so party by using l is equal to proper time. That's a very proper land. Times square root off one minus Reece squared, divided by C square well here and is equal to tree and a tree lighting light years and proper. Lenk is 15 light years into squared rode off one minus. We squared, divided by C square. Well, uh, we have 0.4 is equal to one minus. We squared, divided by C square and three squared, divided by C square Reece squared, divided by C square is equal to zero point mine six and we divided by C equals a 0.98 and therefore we 0.98 times the speed of light. Now let's to won't be We know that s is equal to we multiply by t and solving for g t equals s divided by we Well, hear this tense is 15 light years. 15 light years divided by we is 0.98 times the speed of light and we know they'd One light here is equal to mind. Point for six. Multiply by 10 to the power 15 meters, divided by a 0.98 Multiply by speed of light and therefore Time T is equal to um four. Find it tree multiply by 10 to the power. Eight seconds. Let's go toward this in two years. So 4.83 multiplied by 10 to the power it divided by 3.15 months of life right into the power seven. And this gives us 15 point train tree years. All right now, let's to part see well, a troubled time measured by the astronaut. So travel Time T is equal to three light years, divided by 0.98 times the speed of light tree. Multiply by but one light. Here is 9.46 month of life by 10 to the power 15 leaders and divided by 0.98 Multiply by tree multiplied by 10 to the power eight meter per second and therefore he is equal to nine bind 65 multiplied by 10 to the power seven seconds and it's gonna work this in two years, 9.65 multiplied by 10 to the dollar. Sale in seconds, divided by 3.15 Multiply by. Um, Well, this is three, um 3.15 multiplied by 10 to the power seven and therefore D is equal to 3.6 years.

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