Question

Good day, it there a way it can be proven using calculus that it takes only 42 minutes for an object to go through a tunnel at any place on earth.

Low STAKES WRITING AssIGNMENT 6 (SiMPLe HARMONIC MoTION II - THE GrAVITY TRAIN) Do NOT submit this low stakes assignment! You will use this as part of the background for High Stakes Writing Assignment 2. Topic: Physics was Newton's primary motivation for inventing calculus. With his calculus and his law of gravitation, Newton proved that gravity, that familiar force from everyday life that makes an apple fall to the ground, is also the same force that makes the planets go around the sun and the same force that causes high and low tide on a beach. A strange consequence of Newton's law of gravitation is a hypothetical construction called the gravity train. The gravity train works as follows: imagine linking any two points on the surface of the earth with a straight tunnel through the earth. Using calculus and Newton's law of gravity, one can show that an object (like a shuttle or train) that was dropped down one end of the tunnel would emerge from the other end of the tunnel in 42 minutes, regardless of how far apart the two ends of the tunnel were. This is the basic idea of the gravity train The engineering challenges of building an actual gravity train are significant to say the least However, if the technology existed for constructing these gravity trains, one could travel from New York to Paris or from Connecticut to Alaksa always in 42 minutes The Physics: Let G denote the gravitational constant and let Mand R denote respectively the mass and radius of some arbitrary planet. (Later we can substitute the appropriate values for the earth.) For the sake of simplicity, we will assume that the planet has uniform density and is a perfect sphere. Using Newton's law of gravity (and calculus!), one can show with a good bit of work that the (magnitude) of the acceleration due to gravity at any point P inside the planet i:s GM where 0 sr K R is the distance from the point P to the center of the planet. Moreover, the acceleration due to gravity at P is directed towards the center of the planet. (In particular, this means that the gravitational field at the center of the planet is zero.) For the moment, let's consider two points which are on opposite sides of the planet. In this case, the straight tunne linking the two points would pass through the center of the planet. Now let r(t) denote the position of the train in this tunnel where x-0 denotes the center of the planet Then the ends of the tunnel are located at x R and x--R. For the sake of concreteness let's say the trip in the gravity train begins at r-R at time t 0 and ends at r-R at time t-Tf. Let m denote the mass of the train. Since the tunnel passes through the center of the earth, the force that the train experiences at time t By Newt s second law (i.e., F-ma), we have GMm rt)"(t) 3

Answer 1

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