The solar system is 25,000 light years from the center of our Milky Way galaxy. One light year is the distance light travels in one year at a speed of 3.0 x 10⁸ m/s . Astronomers have determined that the solar system is orbiting the center of the galaxy at a speed of 230 km/s . (b) Our solar system was formed roughly 5 billion years ago. How many orbits has it completed?

Video transcript

Hey, everyone. So this problem is dealing with uniform circular motion. Let's see what it's asking. A planet is located at a distance of 30,000 light years from the center of the Milky Way. Research has confirmed that the planet has a velocity and kilometers per second. When moving about the center milkier determine the number of orbits completed by the planet planet. If it came into existence six billion years ago, it reminds us that one light year equals the distance that light covers in one year at a speed of three times 10 to the eight m per second. And so our multiple choice answers here are a 20 B 21 C 22 or D 2023 orbits. And so the first step here is to recall that our uniform circular motion is in terms of velocity is given by V equals two pi R divided by T or tea, we rearrange for T, it will be two pi R divided by A B and so tea is our period. And so that will tell us how long one orbit takes. And from there, we can figure out how many orbits, we would have, this planet will have made in six billion years. So the trickiest part of this problem, once we have these equations figured out is just unit conversion. So our R or our radius is going to be this distance to 30,000 lights in meters. So we've got three times 10 to the fourth light years and we can multiply that by the speed of light or so that's three times 10 p eight meters per second. And then we have years and seconds. So we need to uh work through those conversions to get seconds into years. So we've got 3600 seconds per hour, multiply it by 24 hours to each per day, multiplied by 365 days per year. And so now all those units canceled and we are left with a radius of orbit of 2.84 times 10 to the 20 m. And then our velocity was given to us in the problem as 190 kilometers per second. So keeping everything in standard units that can be rewritten as 1.9 times 10 to the fifth meters per second. And so from there, we can flood that into tea into our equation. So t equals two, I multiplied by 2.84 times 10 to the 20 meters divided by 1.9 times 10 to the fifth meters per second. And we plug that in and we get 9.4 times 10 to the 15 seconds. So now we're going to want to um, reconvert those seconds back into years. So we can find our period in years. So we'll have one hour or 3600 seconds, one day or 24 hours and then one year for days. And so now we know that tea is 2.98 times 10 to the eight years. And so our last step is to figure out how many rotations we would have in six billion years. And so six billion can be rewritten as six times 10 to the nine years divided by 2.98 times 10 to the eight years. And that equals 20. for approximately 20 rotations. And so that is the answer to this problem. Here we go up and that aligns with answer choice. A so that's all we have for this one. We'll see you in the next video.