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$\bullet$ You take a trip to Pluto and back (round trip 11.5 billion km),traveling at a constant speed (except for the turnaround atPluto) of $45,000 \mathrm{km} / \mathrm{h}$ . (a) How long does the trip take, inhours, from the point of view of a friend on earth? About how many years is this? (b) When you return, what will be the difference between the time on your atomic wristwatch and thetime on your friend's? (Hint: Assume the distance and speedare highly precise, and carry a lot of significant digits in yourcalculation!)

$=0.8 \mathrm{s}$

Physics 101 Mechanics

Chapter 27

Relativity

Gravitation

Rutgers, The State University of New Jersey

University of Washington

Simon Fraser University

McMaster University

Lectures

03:55

In physics, orbital motion is the motion of an object around another object, which is often a star or planet. Orbital motion is affected by the gravity of the central object, as well as by the resistance of deep space (which is negligible at the distances of most orbits in the Solar System).

03:18

Sir Isaac Newton described the law of universal gravitation in his work "Philosophiæ Naturalis Principia Mathematica" (1687). The law states that every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.

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Okay, So in this question, you are in a space shift and you go from dear to the plu to and back. And you know that this total distance that's called the zero is just 11.5 bilion off kilometers. And this ship, the spaceship Hades, a relative velocity through your framed on the earth off 45 1000 kilometers per hour. Okay, in the first I 10 of the problem, we just need to calculate did time that your friends on the earth measure in this journey. So since your friend and on the Earth is in our I knew sure frame he's not under there any relativistic effect. So the time that he measure is just distance divided by relative speed. So this is just 11 0.5 divided by 45 1000. Remember that billions is just 10 to the power off. Nine. So the time that measure is just two point 56 time staying two or a five hours. This is the time. But how many years has passed? Okay, To discover the years, we just need to pick 2.56 times 10 to the bar of five, divided by the number off days in here and the number off hours in a day. So we're gonna get 29 0.2 years. Okay, this is the answer to the first item. The second item we need. Thio, calculate the difference between the time that you measure in your watch and the time that your friend measure in the earth. So we can say that you want to calculate the difference in the time of the earth between the time that you measure. Okay, But how we going to calculate the time that you measure? Well, we know that time is just This says that you see divided by the speed that you move. But the problem here is, since you are under the effects of special relativity, the distance that you see that's called it the desh just Did you think that you see it's going to be lesser than the distance that the rear distance So it's just going to be the d zero divided by gamma because you're experiencing a contract contraction office space. So since we're spirits or contraction of space, you can say the time that you measure is just D zero divided by gamma you and we know that gamma is just Let's simplify desecration. Just great to be d zero Divided by who times one miners. You're square C square in the square root. So this is time that your measure Well, now we can calculate the difference between the two times and this is just going to be which in here time of the earth minus time that you measure is just e zero divided by you and multiplies one miners one minus your squared divided by C square All this in the square root So we just need to substitute all the values in here. We know that the distance is in kilometers so let's put the distance and meters so the distance and meters is just going to be 11.5 time. Stand to the power off 12 now meters, this divided by the speed and the speed. Now we want in meters per second. So we just need to pick 45,000 divided by 3.6. This gonna get just one point 25 time Stand to the power off four. So what we have here one point 25 times 10 4 meters per second. All these the multiply one Miners scrabble her toe one minus one points. 25. I'm Stan for you. Ride it by three times. 10 to the power off eight. Which is Theo the light speed square, square root. And we close here. Okay, so after doing this calculation here, we're going to get That's the difference in time, that's called Delta T is just zero point eight seconds. This is the time differences in the watch and your watch and the watch over your friend.

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