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Problem 56 Hard Difficulty

An alien spaceship traveling 0.600$c$ toward Earth launches a landing craft with an advance guard of purchasing agents. The lander travels in the same direction with a velocity 0.800$c$ relative to the spaceship. As observed on Earth, the spaceship is 0.200 light-years from Earth when the lander is launched. (a) With what velocity is the lander observed to be approaching by observers on Earth? (b) What is the distance to Earth at the time of lander launch, as observed by the aliens on the mother ship? (c) How long does it take the lander to reach Earth as observed by the aliens on the mother ship? (d) If the lander has a mass of $4.00 \times 10^{5} \mathrm{kg}$ , what is its kinetic energy as observed in Earth's reference frame?

Answer

a) 0.946$c$
b) 0.160$l y$
c) 0.114$y r$
d) $7.51 \times 10^{22} J$

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Video Transcript

well, we lost your lander with respect to the earth is equal to We lost your lander with respect to space shitless the lot steals a spaceship with respect to the earth divided by one plus the lost you of lander. With respect to SpaceIy less sorry, it's multiplied by it's multiply by we lost of space. You've been respect to the Earth Divided by C Square. Let's plug into values. Um, 0.800 times the speed of light less are 0.600 times the speed of light, divided by one less. Uh, Zito find hate little zero times the speed of light multiplied by 0.600 times the speed of light divided by C square. And therefore we lost your lander with respect to the earth is zero point 946 times the speed of light. Oh, this was a party. Now let's to part the well, we know it contracted land l A sickle to lp multiply by square Rudolph one minus B squared, divided by C Square. Let's plug in the values with proper length in this case is properly and is 0.200 light years into square root off one minus Zito find 600 hold square. And therefore ah, contract id Land physical to 0.16 theater light years. Okay. No, it's two parts. See? Well, we know it d He is equal to how divided by we the time is distance divided by speed and speed. In this case is a speed off lander with respected e space shitless speed off space, You with respect to the earth and putting the values we have 0.160 lying tears divided by aiding both lost easily him 1.4 words in the times the speed of light. And this is equal to see the point 114 Uh, is there a 0.114 ah, years years. All right. And finally you need to do part D. Well, we know that kinetic energy, uh, kinetic energy is equal to gamma minus one times m c square and we have gone Our bitches one divided by square hood off one minus. We squared, divided by C squared minus one to M C square. No, let's plug in the values. We have one divided by square root off one minus zero point mine for six. We'll scare, um minus one multiplied by. Well, I am in this case is for multiply by 10 to the power five and speed off. Mind his three year old to buy me 10 to the power it holds square and therefore kinetic energy is equal to 7.5 run multiply by 10 to the power 22 chills.

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