The gravitational attraction between two objects with masses m...

Question

The gravitational attraction between two objects with masses m_A and m_B, separated by distance 𝓍, is F = Gm_Am_B/𝓍², where G is the gravitational constant. If one mass is much greater than the other, the larger mass stays essentially at rest while the smaller mass moves toward it. Suppose a 1.5 x 10¹³ kg comet is passing the orbit of Mars, heading straight for the sun at a speed of 3.5 x 10⁴ m/s. What will its speed be when it crosses the orbit of Mercury? Astronomical data are given in the tables at the back of the book, and G = 6.67 x 10^-11 Nm²/kg².

Accepted Answer

Hey, everyone. So this problem is dealing with work and gravitational forces. Let's see what it's asking us. We have Newton's law of universal gravitation gives the gravitational force of attraction between two objects with mass as the force is equal to G, the gravitational constant multiplied by M one multiplied by M two, all divided by R squared using Newton's second law. If one object is more massive, the massive object remains at rest while the lighter object moves towards it. So now we have a space boulder with a mass of 1.2 times 10 to the 9 kg passing Jupiter's orbit directly towards the sun at a speed of 45 kilometers per second. And we're asked to determine the speed of the boulder when it reaches the earth's orbit. We're told that we can use any necessary astronomical data from literature sources. We can look up other constants. Our multiple choice answers here are a 9.27 times 10 to the third meters per second. B 6.43 times 10 to the fourth meters per second. C 5.88 times 10 to the fourth meters per second or D will not make it to the earth's orbit. OK. So the first step we're going to take here is using the equation they give us in this problem. So we have the force is equal to gr gravitational constant multiplied by M one multiplied by M two, the masses of the two objects all divided by R squared where R is the distance between those two objects. So the speeds here are changing the kinetic energy then is changing. And so the work energy theorem states that the network is equal to the change in kinetic energy. And so our work is going to be equal to the integral of our force multiplied by the derivative of our radius or as our radius is changing. And we'll take that integral from our initial radius to our final radius. And so then we can plug in that equation for the force. So we have GM one M two all divided by R squared multiplied by Dr We take an integral of that from our initial radius to our final radius. Recognizing that during this problem, the gravitational constant and the masses do not change, we can pull that out of our integral. So here and our equation becomes work is equal to GM one M two multiplied by the integral from our initial radius to our final radius of Dr divided by R squared. And so we're calling our derivative rules. Oh Sorry. Our integra sorry. Our integration rules when we integrate dr divided by R squared, we are left with negative one divided by R. And so we'll multiply that by our constant GM one and two. And then that negative one divided by R is going to be from our initial radius to our final radius. And I can further simplify this out where we have GM one M two multiplied by negative one divided by RF minus a negative. So plus one divided by an R I. And so now what we need to, so now we have the problem to the point where all we need to do is figure out what each of these terms is. And so what we have here, it's what we need to find our work, our network. And as we said earlier, we can recall that our change in kinetic energy is equal to our net work. Our change in kinetic energy is important because that's where we're going to find our seed truck. The speed of the boulder is what we're ultimately looking for. And so our change in kinetic energy is of course, just our final kinetic energy minus our initial kinetic energy. And we can recall that the equation for kinetic energy is one half M squared. So this will be one half and BF squared minus one half M the I squared. Because the boulder is moving in this scenario, it's going to be the mass and the speed of the boulder that we are looking for. And so now we can put these two equations together and we come up what our work, which is G multiplied by M one multiplied by M two multiplied by the quantity negative one divided by RF plus one, divided by R I is equal to one half MBVFB squared minus one half MBVIB squared. And so let's take a look at each of these terms. So being able to look up our astronomical data, our m one is going to be the mass of the sun. And we know that that is 1.99 times 10 to the kg and two is the mass of the boulder, which the problem tells us is 1.2 times 10 to the 9 kg. Our final radius is going to be the distance from the earth to the sun because we're looking at the boulder when it enters the earth's orbit. And so we can look that up and that is 1. times 10 to the 11 m. Our initial radius is the distance from Jupiter to the sun. And that is when we look that up 7.78 times 10 to the m. And then from the problem, we are told that that initial speed of a boulder as it's passing, Jupiter is 4.5 times 10 to the fourth meters per second. And then our gravitational constant, which we could look up. It was also given to us in the problem. 6.67 times 10, the negative 11 m cubed per kilograms second squared. And so from here it is just a pretty lengthy but doable plug and shot problem. So let's see what we have here. So G 6.67 times 10 is the negative 11 m cubed per kilogram second squared multiply by and one that's the mass of our sun 1.99 times 10 to the 30 kg multiplied by M two, the mass of our boulder 1.2 times 10 to the 9 kg multiplied by negative one divided by RF 1.5 times 10 to the m plus one over R I 7.78 times 10 of the 11 m. All of that is equal to the right hand side of our equation which is one half multiplied by the mass of the boulder. So 1.2 times 10 to the 9 kg multiplied by our final seat of our boul boulder. So VFB squared minus one half multiplied by 1.2 times 10 to the 9 kg multiplied by RP I of the boulder 4.5 times 10 to 4 m per second. And that quantity squared. And so we can take a few steps to simplify the math in this problem, to solve each of these terms out to make this a little bit more manageable as we isolate how our speed. And so that leaves us with 8. times 10 to the 17 jos is equal to six times 10, 8 kilograms multiplied by BFB squared minus one point times 10 to the 18 jolt. And so that's a little bit more manageable to isolate our VB. Sorry, our BF of the boulder term. And so that looks like our vinyl feet of our boulder is equal to, I swear it of 8.571 times 10 to 17 jewels plus 1.215 times 10 to the Jews all divided by six times 10 to the 8 kg. Plug that into our calculator and we get 5. times 10 to the 4 m per second. And so when we take that answer, we look at our multiple choice answers because we know we chuck a step to simplify the math to make our numbers a little bit easier to work with. We can be confident in choosing answer choice C which is 5.88 times 10 to the 4 m per second. And so answer choice C is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.

Key Concepts

Gravitational Force

Conservation of Energy

Kinetic and Potential Energy

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