text: Suppose you dug a tunnel through the center of Earth, jumped in, and let gravity pull you through. How long would it take you to reach the other side of the planet? For decades, physics students have been asked to calculate that time and have been taught that the correct answer is 42 minutes. Now, a more realistic analysis has lopped 4 minutes off that estimate. 
   
   "This is the kind of paper we love," says David Jackson, a physicist at Dickinson College in Carlisle, Pennsylvania, and editor of the American Journal of Physics, a publication of the American Association of Physics Teachers. The new calculation, which appears in the March issue of the journal, doesn't merely add more detail about the structure of Earth, he notes. It also explains why you can replace one overly simple assumption with another equally crude one and still get a much more accurate answer. "That's what makes this so much fun to think about," Jackson says. 
   
   The gravity tunnel problem is a staple of introductory physics courses because it simultaneously demonstrates both striking features of Isaac Newton's law of gravity and a common but very important type of cyclical motion. To solve it, students must calculate how the force of gravity on an object changes as the thing falls through the tunnel. 
   
   Here's where the usual unrealistic assumption enters. Students assume that, like a billiard ball, Earth has the same density throughout: roughly 5500 kilograms per cubic meter. In that case, the strength of the gravitational force pulling you toward Earth's center varies in proportion to the distance from the center. That's because as you descend through the tunnel, the amount of mass at an altitude lower than yours decreases, whereas the mass at an altitude higher than yours no longer has any effect on you—as students calculate using a bit of mathematics called the shell theorem. 
   
   Because the force pulling you toward the center is proportional to the distance from the center, you zip back and forth through Earth exactly as a weight on a spring bobs up or down or a pendulum swings back and forth. The gravity tunnel problem is used in teaching precisely because it produces such simple harmonic motion. 
   
   In reality, of course, Earth does not have a uniform density, but has a less dense crust and mantle and a more dense core. So Alexander Klotz, a graduate student in physics at McGill University in Montreal, Canada, started thinking about what a more realistic analysis would yield. Klotz says he isn't sure why he started mulling over the problem. But he sometimes answers physics questions on the website reddit. "I've sort of engaged in a bit of educational outreach," Klotz says, "and that [question] comes up quite a lot." 
   
   To get a more realistic mass distribution for Earth, he relied on the Preliminary Reference Earth Model, which is based on seismic data. It traces Earth's density from about less than 1000 kilograms per cubic meter at the surface to roughly 13,000 kilograms per cubic meter at the center of the core 6371 kilometers below, including a dramatic jump at the edge of the outer core, 3500 kilometers from the center. Solving the problem numerically, Klotz found that an object should fall through Earth in 38 minutes and 11 seconds, instead of the 42 minutes and 12 seconds predicted assuming a uniform planet. 
   
   Curiously, Klotz found that he got almost exactly the same answer—38 minutes flat—if he simply assumed that the force of gravity remained constant and equal to the value at the surface as an object plummeted toward the center. Such a constant force would require a different density distribution, one that increases steadily as the distance to Earth’s center falls—so that when the distance to the center is halved, the density doubles—and peaks to infinity at the center. (The real distribution simply plateaus in the core, and the force of gravity diminishes to zero at the center of the core.) 
   
   So why does the constant-force approximation work so well? Because of the mass distribution in Earth, the force of gravity remains roughly constant—and actually increases slightly—down to the outer core, Klotz explains. From there on in, the force of gravity falls with distance much as in the original problem. But by that point, the object is moving so fast that it actually spends very little time passing through the core, where the constant-force approximation is obviously wrong. So in the end, the approximation is good enough. 
   
   It's that surprising explanation that makes the new analysis gratifying, Jackson says. "The classic problem needs to stay," he says. "But this is a nice addition to the classic problem." For his part, Klotz says that, in this day of big science, his experience shows that "with the right idea it's still possible to make, not a monumental discovery, but an incremental one." ||||| Abstract 
   
   This paper examines the gravity tunnel using the internal structure of Earth as ascertained from seismic data. Numerically, it is found that the time taken to fall along the diameter is 38 min, compared to 42 min for a planet with uniform density. The time taken to fall along a straight line between any two points is no longer independent of distance but interpolates between 42 min for short trips and 38 min for long trips. The brachistochrone path (minimizing the time between any two points) is similar in shape to the uniform-density solution but tends to reach a greater maximum depth and takes less time to traverse. Although the assumption of uniform density works well in many cases, the simpler assumption of a constant gravitational field serves as a better approximation to the true results. ||||| A composite image of the Western hemisphere of the Earth. Credit: NASA (Phys.org)—Alexander Klotz a student at McGill University in Canada has calculated a new answer to the commonly asked physics question, how long would it take a person to fall all the way through the Earth? Instead of the commonly accepted 42 minutes, he claims it is 38. He has published his reasoning, math and conclusions in a paper published in The American Journal of Physics. 
   
   If someone were to drill a hole all the way through the planet, and then somehow manage to fall into it, how long would it take them to arrive on the other side? That is a physics question put to students every year, and those who give it expect the answer to be 42 minutes. But is that answer correct? Klotz says no and has the math to prove it, Science reported. 
   
   The accepted answer of 42 minutes takes into account the constantly changing impact that gravity will have (and ignoring drag due to the presence of air) on the person falling, becoming less and less of a factor as the center of the Earth is approached then growing stronger and stronger as the person heads "up" against gravity on the other side. It is accepted that the speed attained during the descent on the first half of the journey would be significant enough to cause the person to continue moving against gravity on the other side of the planet, right up until the surface is reached. 
   
   But Klotz argues that it is time to start taking the different densities of the Earth's layers into consideration—after all, a lot of research has shown that our planet is a lot denser at the center than at the crust for example—and that of course would have an impact on the person falling through. He has used seismic data to calculate the different densities at different depths and has used that data to give a more accurate answer to the falling man question, stating that it would in fact, take just 38 minutes (and 11 seconds) to fall all the way through, not 42 and (12 seconds). 
   
   Interestingly, Klotz also notes that if gravity were to be assumed to be at a surface level constant throughout the duration of the trip, the math shows it would take just 38 minutes as well. 
   
   Explore further: Are astronauts really weightless? 
   
   More information: The gravity tunnel in a non-uniform Earth, Am. J. Phys. 83, 231 (2015); dx.doi.org/10.1119/1.4898780 . On Arxiv: arxiv.org/abs/1308.1342 
   
   Abstract 
   
   This paper examines the gravity tunnel using the internal structure of Earth as ascertained from seismic data. Numerically, it is found that the time taken to fall along the diameter is 38 min, compared to 42 min for a planet with uniform density. The time taken to fall along a straight line between any two points is no longer independent of distance but interpolates between 42 min for short trips and 38 min for long trips. The brachistochrone path (minimizing the time between any two points) is similar in shape to the uniform-density solution but tends to reach a greater maximum depth and takes less time to traverse. Although the assumption of uniform density works well in many cases, the simpler assumption of a constant gravitational field serves as a better approximation to the true results. 
   
  
target: – It's a common hypothetical in physics classrooms: How long would it take to fall through a tunnel from one side of the Earth to the other? The new answer: 38 minutes and 11 seconds, reports Science. The result is courtesy of McGill University graduate student Alexander Klotz, whose calculations take four minutes off the previously accepted answer and appear in the American Journal of Physics. The reason for the shorter time? Unlike previous calculations, Klotz's take into account that the Earth's layers have different densities, and those differences would affect the hypothetical falling man, notes Phys.org. "The way the Earth is structured, the gravity increases slightly as you go deeper towards the dense core, to about 110% of its surface value, before getting weaker as you move through the core, reaching zero at the center,” says Klotz, as quoted in Forbes. (If it's all a little too hypothetical, try physics-inspired pasta instead.)