How long would it take to fall through the center of the Earth?
Although many cartoons suggest that if you dig a deep hole straight down through the Earth, you will end up in China. The truth is that starting a hole in North America will take you to the Indian Ocean. In order to emerge in China, you would need to start digging in Argentina. Having clarified this technical point, we are still stuck with the figure of speech that digging through the Earth form the U.S will take you to China. The interesting part is that falling from here to the Indian Ocean would take as much time as falling from here to China. In other words, as long as you are falling though some hole in the earth, it will take the same amount of time to fall. Let’s investigate this seemingly paradoxical statement.
Let’s assume that we have a hole that leads straight through the center of the Earth and is large enough to comfortably accommodate you while flailing your arms. To keep the calculations simple, we will neglect air resistance.
The force of gravity between two masses m and M is
In this problem, you are the m and the Earth is M. The big G as it is often referred to is the gravitational constant equal to
The letter r is the distance from you to what’s pulling you. A neat fact is that for something like the earth you can pretend that all of the mass between you and the center of the earth is pulling you from the center of the earth. But when you start falling, things start changing. As you fall there is less and less mass between you and the center so there is less and less to pull you down. But wait! You are closer to the center and the closer you are to what’s pulling you the harder you will be pulled. Acceleration tells you how fast your speed is changing, or how quickly you are falling more quickly. Newton came up with a mathematical way to relate a force, such as gravity, and acceleration, F=ma. In the case of falling through the earth it is gravity that is causing the acceleration so we say that gravity is the force in this equation. When you are on the surface of the earth the acceleration due to gravity is 9.8 m/s2. The acceleration due to gravity depends on the mass that’s pulling so once you being to fall, your acceleration gets smaller.The mathy way to write this is
But what happened to your mass in this equation? Wouldn’t your acceleration depend on whether or not you had that 1 lb hamburger for lunch? Turns out that anyone from a newborn baby to the Incredible Hulk would accelerate the same if they fell through the hole. This is was exactly what Galileo famously demonstrated when he dropped various masses from the top of the leaning tower of Pisa. Furthermore, this also shows us that mass which creates gravity is the same stuff that gives you inertia. This last tidbit is what made Galileo famous.
Since the mass M and the distance r are both changing, there are too many variables. However, there is a clever relationship between mass and distance. It’s called density. Density is the amount of mass of an object divided by the volume occupied by that object p=M/V. In this case, we are talking about the density of dirt and rock. But there is a “v” in the density equation, not a distance. Remember that the volume of a ball is
and you can see where the distance comes in. We are going to assume that the all of the dirt, rock and magma in the earth has the same density. A geologist may disagree but our only other choice would be to discuss differential calculus. At this point, given the choice of whether to solve nonlinear differential equations or upset a geologist, we will choose the latter. We can rearrange the density equation to say that the mass is density times volume M= ?V. And the density of the Earth is the total mass of the Earth M divided by its volume
With this information, the force becomes
Now this is still a bit bulky. Let’s see if we can use our noodle to expunge a few more variables. We mentioned earlier that gravity on the Earth’s surface is
It just so happens that that expression is hidden in the force equation. This leaves us with the force
So now we know what will happen on your way to the center, but what happens when you go flying through the middle of the earth? Now you have gravity trying to pull you back toward the center and its slowing you down as fast as it sped you up. And because it is making you slower at the same rate it made you faster, you will stop just as you pop out of the other side of the hole. And then the whole thing starts all over again. Imagine a native on the surface awaiting your brief visit. They would see you squirt out of the hole to a height equal to that of which you jumped into it. They would also hear screaming: high pitch and then low as you descend. The reason for your screaming will become clear once we do the calculations.
Now how long would it take to fall through the Earth? The time of one oscillation, which is the time for one round trip through the hole is called the period. Solving for the period of a gravity hole from scratch required some hefty mathematics like Calculus. Luckily Isaac Newton wrote the first book on Calculus called The Principia. And fortunately for us, he gave us a way to find the period. In the case of the gravity hole, the period of your falling oscillation is
The period is a around trip and hence twice the amount of time it takes to travel through the Earth. So we will need to divide the period by 2. Before we plug in the numbers, notice that it doesn’t matter how heavy you are. Big or little, you it takes the same amount of time to fall. In fact you will feel weightless as you fall.
The acceleration of gravity is 9.8m/s2 and the radius of the Earth is 6.378 million meters. This means that you would fall through the entire Earth in only 42 minutes! Can you imagine traveling 8 thousand miles in less than an hour? You maximum velocity at the center would be roughly 8km/s (18,000 mph). That’s fast enough to make you scream.
What about air resistance? Air complicates the problem tremendously. It makes the equations very difficult to solve and requires a few broad assumptions about the falling object. For instance, the air will travel differently around a large person than a kid. However, air resistance leads to an interesting phenomenon: terminal velocity. The air act like a frictional force and counter acts the acceleration of due to gravity. Without air, an accelerating object will continue to increase its velocity. With air, the frictional drag force increases as the object moves faster. At some velocity, the frictional drag force becomes equally opposed to the gravitational force. The object then coasts at that velocity. This is what is meant by terminal velocity. A typical raindrop falls at about 9m/s while the average person has a terminal velocity of 56m/s. Assuming that you are traveling at this terminal velocity for the duration of your fall, you will arrive in the Indian Ocean after only two and a half days. Be sure to pack a lunch and your goggles!