The Escape Velocity of the Earth

Topher Cawlfield
March 19, 1996

Abstract

The problem of computing the escape velocity of the Earth away from itself, at sea level, is considered. A quick logical analysis shows the question to be insoluble, and very silly. The solution is then obtained through a series of perpendicular arguments and assertions. The escape velocity of the Earth is found to equal the speed of sound in a black hole.

Introduction

How fast must one throw the Earth upward so that it never returns to the ground? How quickly must the Earth move to escape it's own gravitational well? This seemingly academic question is a problematic one for most logicians. Yet the question is an important one -- we don't want to loose the Earth by accident, do we? The escape velocity of the Earth must be an important consideration in the construction of environmentally friendly experiments on our little planet.

Logical arguments do not solve this problem. A perpendicular approach is necessary. We first show that clouds do not fall to the ground. Well, not as fast as dead birds, anyway. We then examine earthquakes, and have ourselves a strong cup of coffee. After a short bath, we are almost finished. The entire problem is solved pending a quick coin toss. Probably.

Parallel Logic

Let us say, for example, that one manages to push the Earth at a rate of 5 m/sec straight upwards. A subsequent measurement would show that the Earth is moving 5 m/sec straight upwards, keeping an even pace with itself. We find that the Earth's gravitational field, anchored to the Earth by the laws of gravity, closely track our little planet's every movement. A harder push is of no avail. Limited experimental evidence supports this notion.

One must then come to the conclusion that there is no escape velocity for the Earth. But everything else has an escape velocity. In fact, everything else has the exact same escape velocity! So why not the Earth? The problem lies in our narrowed view of logic.

A Perpendicular Solution

First, examine things which do not fall. Clouds, for example. By the cloud's example we see that gravity is not all it's cracked up to be. Neither is the ground. Earthquakes can occur when the ground becomes cracked. By this mechanism the ground can separate from itself. Therefore we must insist that the Earth can leave the ground. But at what tax rate?

Consider drool. Now, consider a common ping-pong ball, bouncing on the ground during an earthquake. The ball will weigh 2.5 grams. This fact may become important later on.

By neglecting atmosphere, one can prove that the escape velocity of an item, such as a fence, is reached just before it is demolished by the cruel ground. Given that you dropped it from semi-infinite hight, at least. This is probably due to a lack of oxygen to the brain. Atmosphere makes one's thinking much cloudier, and hence makes the problem more complex. But since the atmosphere is arguably a part of the Earth, I will neglect it. I will neglect any other facts I choose to as well.

What would happen if an alien race attempted to abduct a garbage dump? Would this be immoral? Or do we owe them money for the favor? Our money would not be as valuable to them as the garbage, or else they would rob a bank instead. So they wouldn't mind paying us for giving them our trash. We would be wise to produce as much garbage as we can before aliens come.

Our job is now to decide how fast the ground can hit itself. Recall our study of earthquakes. It should be obvious that the speed of an earthquake, dropped from an infinite height, will equal the speed of sound in a black hole. This is left as an exercise for the reader. The following formula may help:

By reasoning backwards and a little to the left, we find that the escape velocity of the Earth from itself is equal to the speed of sound inside a black hole minus a toothpick. We can almost neglect the speed of the toothpick, since it is so large compared to a singularity.

Conclusion

We thus conclude that the velocity of the Earth, when escaping sound in a hole, is black. This should appeal to your intuition so much that you may feel you've wasted your time reading this. I assure you, this is not the case! You have instead wasted someone else's time. Further analysis could explore such questions as "how fast does sound travel in a black hole?" And "who's time have I wasted really?" Apparently paperweights don't function as well as they should in a hurricane, which demonstrates the inherent value of thinking perpendicularly logically.