If I fire a gun while near light speed, will the bullet EXCEED light speed?

Dear Cecil:

I have a question about the inability of matter to exceed the speed of light. Suppose I am in a spaceship traveling at the speed of light minus five miles per hour. What would happen if I fired a gun in the direction that the ship is moving? Or tried to run forward at six miles per hour? Would I prove Einstein wrong?

Cecil replies:

There are two ways we can go about this. First there’s the Way of the Wimp, wherein I simply tell you no, you wouldn’t prove Einstein wrong, and we leave it at that. This avoids distracting complications, but leaves something to be desired from the standpoint of intellectual rigor. Then there’s the Way of Righteousness, which requires mental tenacity and moxie. Fortify yourself and we’ll give it a shot.

You probably have the idea that if you’re standing in a bus moving at speed u, and you walk forward at speed v, your total forward speed w is expressed by the straightforward sum u + v. Alas, this is a cruel illusion. In reality, what we might call “addition of velocities” is governed by the awe-inspiring equation

w = (u + v)/(1 + uv/c²)

where c² is the speed of light squared. (This may give you pause next time you hike to the can on a Greyhound.) At so-called Newtonian (i.e., slow) speeds, the term uv/c² is pretty close to 0, and the equation reduces down to the familiar w = u + v. However, if we are traveling at, say, 0.9c (nine-tenths the speed of light), and we shoot a bullet forward also at 0.9c, we discover via the above formula that the slug doesn’t attain an overall speed of 1.8c (i.e., more than the speed of light), but rather a modest

(0.9c + 0.9c)/(1 + [0.9]²) = 0.994c


Does this mean the bullet just dribbles out of the gun like a freaking gumdrop, for Chrissake? Not at all — to you, the space traveler, everything looks normal. However, a stationary observer would note that you were suffering from the unique effects of the Fitzgerald contraction — which is to say, (1) time would slow down for you (although you wouldn’t realize it), and (2) you and your spaceship would get compressed like an accordion along your axis of travel.

The following poem may help to illustrate this:

There once was a racer named Fisk
Who took a considerable risk.
When his dragster got traction
The Fitzgerald contraction
Reduced his wazoo to a disc.

Sorry, couldn’t resist. Anyway, if you ponder this matter awhile, you’ll inevitably come to the following conclusion: the faster you go, the slower you go. Ergo, the speed of light cannot be exceeded.

And you wonder why physics is my favorite subject.

Send questions to Cecil via cecil@straightdope.com.

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