How long does it take to fall through the center of the Earth to the other half vertically?

Suppose you dig a long tunnel to the center of the Earth, jump in it and let gravity pull you down. How long will it take you to reach the other side of the globe? Every year physics students have to study this lesson and the correct answer is 42 minutes . Now, analyzing the fact that the correct answer is actually four minutes shorter.

"This is a kind of problem that everyone is interested in," said David Jackson, a physicist at Dickinson University in Carlise, Pennsylvania and editor of the American Physics Journal, published by the American Physical Teachers Association. The new solution printed in the March journal not only changes much about constants that represent the earth structure, but also explains why if you change a basic assumption in the problem with another assumption. still give the correct answer. "That's why the problem is so fascinating , " Jackson said.

Gravity tunnel problem is a set of basic problems of physics because it both demonstrates the essential rules of Newton's law of gravity and the law of cyclic motion. To solve this problem, students must calculate how gravity affects objects that change according to the process of falling in the tunnel.

Earth has a uniform density: about 5500kg per cubic meter.

An unrealistic assumption is applied, that is like a billard ball, Earth has a uniform density: about 5500kg per cubic meter. In that ideal case, the extent to which gravity impacts you changes in proportion to the distance to the center of the earth. That is because when you fall deeper and deeper into the tunnel, the volume of matter below becomes lower and the pressure from the mass of matter above decreases.

Because the force pulling you to the center is proportional to the distance to the center of the Earth, you will move forward and backward between the two ends of the tunnel like a pendulum that moves left - right. Gravity tunnel problem is used in teaching because it is a good example of simple cyclic movement .

In fact, it is clear that the Earth does not have such uniform density of material , but has a low density shell and high density core.

Alexander Klotz, a year-long student who graduated in physics at McGill University, Montreal, Canada, began to look for a more realistic analytical result. Klotz said it was a common question on the science section of Reddit.

For more realistic data on Earth's material density, Klotz used the Earth's preliminary reference model, based on seismic data. As a result, the material density at the surface is about 1 ton per cubic meter and about 13 tons per cubic meter at the position of 6.371km below ground.After the calculation, the results showed that an object would take 38 minutes and 11 seconds to fall through the real tunnel, compared to 42 minutes and 12 seconds with the ideal tunnel.

The density distribution of the Earth makes gravity actually not change much.

Strangely, Klotz found that the answer remains the same if we let gravity be a constant, unchanged during the falling process. In order to obtain constant force, the material density must be specially arranged so that when the distance to the center decreases by half, the material density doubles and reaches infinity at the center (in fact gravity drops to zero at the center and the density at the center does not reach infinity).

So why if we think that gravity doesn't change, is the answer still correct? That's because the Earth's density distribution makes gravity really not much - just a little higher - until the core position. After passing through that position, the gravity decreases like the original one. The problem is that at that time, the object will have a tremendous velocity that passes through the Earth's core in the blink of an eye, where the assumption of constant gravity becomes absurd. So even if there is constant gravity, the answer is still the same.

This is what makes the new prize interesting. The old solution cannot be dismissed, but this solution is an interesting additional knowledge for the problem.