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Problem 43 Hard Difficulty

An astronaut wishes to visit the Andromeda galaxy, making a one - way trip that will take 30.0 years in the spaceship’s frame of reference. Assume the galaxy is 2.00 million light - years away and his speed is constant. (a) How fast must he travel relative to Earth? (b) What will be the kinetic energy of his spacecraft, which has mass of $1.00 \times 10^{6} \mathrm{kg}?$ (c) What is the cost of this energy if it is purchased at a typical consumer price for electric energy, 13.0 cents per kWh? The following approximation will prove useful:
$$\frac{1}{\sqrt{1+x}} \approx 1-\frac{x}{2} \quad \text { for } x<<1$$

Answer

a) $1-1.13 \times 10^{-10}$
b) $5.99 \times 10^{27} J$
c) $\$ 2.16 \times 10^{20}$

Discussion

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AA

Alexis A.

February 13, 2021

on part B, Why are does the exponent of (v/c) is - 12 and not -10 like in part 10?

Video Transcript

but let's to part a. Well, we have distance. D is equal to a 2.0 multiply by 10 to the power six light years on This equals 2.0 multiplied by 10 to the power six, uh, years times the speed of light See? And we know they Delta t is equal to calm a times proper time proper time. And this equals, uh, 30 years divided by square root off one minus one minus We squared, divided by C square We'll hear appropriate Time is 30 years Onda We know that we speed equals distance divided by time on DDE Therefore we have we is equal to, um 2.0 multiplied by 10 to the power six years times the speed of light divided by square root off Um um in fact, a 30 years divided by a squared Rudolph one minus Oh, we divided by sea Whole square. So 30 years divided by square root of one minus Really? Why did they see hold square And and this implies state, um we divided by sea is equal to one divided by squared Rudolph one place 2.25 multiply by 10 to the power minus 10 on Dhe. Further, we divided by C equals ah one minus 1.1 free. Multiply by 10 to the power minus 10. All right, now, let's do part of B Well, to part B. Well, we know that Grandma is equal to one divided by one minus. We divided by see whole square. And therefore gumiho is equal to one divided by one minus. Um, bracket clothes on minus one point. Since we divided by seas, um, one minus one point 13 Multiply by 10 to the power minus 12. And we have square and therefore God, my physical to 66519 Now we know that kinetic energy, uh, is equal to gamma. Minus one times M C square. And now well, we have gamma, which is 66519 minus one, multiplied by M C square. Well, M C square m is a one multiplied by 10 to the power uh, six. Multiply by three. Multiply by 10 to the power eight whole square. And so in this kinetic energy, eyes equal to 5.99 five points. 99 multiply by 10 to be bothered. 27. Jules. All right. No, let's do Marcie. Well, In boxing, we need to find out. Cost Well, calls physical too. 5.99 Multiply by 10 to the power 27 27 divided by are 3.60 multiplied by 10 to the power six multiplied by, um, this a 0.13 dollars. Um, you are killem. What are insulting this weekend? Course physical too. But two point 2.16 multiplied by 10 to the power 20 dollars.

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