 Let's jump into an example. Humanity has just dispatched its first spaceship on its way to Jupiter to examine a mysterious alien monolith, while the ship is somewhere between Jupiter and the Sun. The ship's computer notices that the accelerometer measures zero acceleration. Given that the engines were turned off, how far was the ship from Jupiter? Let's unpack this question. We know that the engines were turned off, which means any force on the ship is solely the result of the gravitational field from Jupiter and the Sun, assuming that the other planets are really far away. And since there's no acceleration at all, the total field strength is zero. This means we're looking for the point between the Sun and Jupiter where Jupiter's gravitational field is equal strength but opposite direction to the Sun's gravitational field. This is actually pretty similar to finding the point where the two gravitational forces are equal, but without the need to worry about the mass of the spaceship. Now on a line between the Sun and Jupiter, the fields are already pointing directly away from one another, so we just need to concern ourselves with the field strength. As we've discussed, the field strength of an object is equal to the universal constant of gravitation multiplied by the mass divided by the distance squared. So let's set the field strength of the Sun equal to the field strength of Jupiter. Now RS and RJ are the distance from the spaceship to the Sun and to Jupiter respectively. To find RJ, which is what the question has asked for, we can use the distance from Jupiter to the Sun to remove RS. Substituting this in, we find the following. We can then rearrange this equation to find RJ. Firstly, we divide both sides by big G, then we rearrange to put the radiuses on one side and the masses on the other, and then we take the square root of both sides. Next, we divide the left-hand side by the distance to Jupiter and move the one to the other side. Finally, we rearrange our equations that we have an expression for the distance to Jupiter. If we substitute in our values, we find that using the positive root of our equation, the distance from Jupiter is 2.446 times 10 to the 7 km. Using the negative root in our equation, we find that the distance from Jupiter is equal to minus 2.602 times 10 to the 7 km. And there we have it. There are two answers because there are two points where the strength of Jupiter's gravitational field is the same as the Sun's, one on each side of Jupiter. Since we're only interested in the solution, where the fields point in opposite directions, and where the spaceship is travelling between the Sun and Jupiter, then we know that we need to select our first answer, where our distance from Jupiter is equal to 2.446 times 10 to the 7 km.