To Infinity...

 
Image from  Wikipedia

Image from Wikipedia

 

The time has come to post some real material to this site—a simple exploration of physics that will (hopefully) build audience interest and help me learn techniques of generating math-laden web content. After spending the last couple of days mulling over which one of my little projects would be the first one I share, the answer came to me late last night in the form of a text message from a friend:

 
Technical question. Who to ask? You, of course. Escape velocity from Earth is about 7 miles per second. But that is for orbital velocity. What if you just aim straight up from the ground? No orbiting, just straight up. In that case will 7 miles per second, straight up, allow you, or me, or both of us to keep going, and going, and going, and.....
— JB
 

Excellent question! And, the perfect metaphor for getting this website off the ground. Thanks, JB!

The answer is yes, and we shall examine the formula for planetary escape velocity (\(v_e\)) to see why:

\[\begin{equation}
v_e=\sqrt{\frac{2 G M}{R}}
\label{080717:ve}
\end{equation}.\]

\(M\) and \(R\) represent our planet's mass and radius, respectively. \(G\) is the gravitational constant.

Using Earth's values, one finds that

\[v_e=11.186\text{km}/\text{s}.\]

(For residents of Liberia, Myanmar, and the United States of America, that's \(6.951\text{mi}/\text{s}\).)

Now, here comes the fun part: we can learn just as much about \(v_e\) by noting what \eqref{080717:ve} does not contain, i.e. references to anything that might be escaping. The object's mass, it's size and shape, it's direction of travel—none of that enters the equation. It is the environment (planet) that determines \(v_e\), and as long as an object's speed is greater than \(v_e\), it will leave that environment and never return!

Finally, there is one more subtle (and unfortunately necessary) point to be made: the escape "velocity," \(v_e\), is actually an escape speed.

In physics, speed and velocity have two different meanings. Speed describes the distance an object travels within some amount of time (or in other words, how fast it's going), while velocity is the combination of an object's speed and the direction in which it is moving. Speed and velocity are related, but they are different! Nevertheless, the terms are often used interchangeably, even amongst physicists, which is how we wind up with things such as \(v_e\) being called "escape velocity" (even though it's a speed!)

That point, the difference between velocity and speed, cannot be made enough. Why? Because knowing this difference allows one to easily recognize that \eqref{080717:ve}, which represents speed, says absolutely nothing about the direction of the object's motion. Therefore, it does not matter whether the object is moving straight up and away from the planet's surface, off towards the horizon, or anywhere else in between; if its speed is greater than \(v_e\), then it is not coming back.

To summarize, once again, the answer to the original question is yes—no matter what the object is and in which direction it is traveling, having a speed greater than \(7\text{mi}/\text{s}\) means it leaves the Earth forever.*

Ad Astra!

Aaron

*This statement assumes that the planet's gravity is the only force acting upon the object, and that said object does not hit anything on its way out—the ground, the air, etc. Why these assumptions are necessary is a post for another day!