Tag Archives: Train

A short post on length contraction

20 Mar

Here’s a quick recap of the things we know about relativity; the perception of time is dependent upon velocity, and the speed of light is a constant regardless of the frame.(check out the full episode #1 here and #2 here). Alright, so we know something about the time and we know something about the speed. If you’ve ever dealt with rates before, you might be wondering “What’s happening to the distance?”

That’s a fair question, but it’s a little more complicated than it appears on the surface. Let’s think back to our train problem. If we want to measure the train at rest, we can hold a tape measure up to the ship and record the length. That’s works because the ship and the tape are both at rest with respect to one another, but what if we wanted to measure the space ship while it’s moving?

Now we have a problem. We could hold the tape measure still and wait for the train, but we’d have to be able to make a length measurement instantaneously as the train reached the tape. It’s not a good plan. We could move the tape measure at the same speed as the train, but then we’re taking another measurement with the tape at rest with respect to the train. Hmm. Let’s try something different. Bring in the timer gates.

image095

Here’s a quick refresher on timer gates. They are a gate with a laser between them (here represented by the dashed lines). When an object passes through the gate and breaks the laser, an electrical impulse is sent down the black wire to the computer which records the time. That’s all there is to it.

How can we use these? If we set up two along the tracks of the moving train, we can use one to record when the back of the train goes by and the other to record the arrival of the front of the train. You could do some math to determine the length of the moving train as a function of the time difference and the distance between the gates…but you probably don’t like math. The other option is moving the gates until the back of the train goes by one gate at the exact time that the front of the train arrives at the other.  At that point, the distance between the gates is the length of the moving train.

Alright, so I do that experiment and I find that the length of the moving train is related to the resting length of the train by this equation

image009This means that the moving object appears to be shorter than the object at rest by some factor related to the speed at which it’s moving. Actually, that factor seems kind of familiar. Didn’t we see that when we were dealing with time contraction?image008Yep. Same factor. That’s because it needs to be in order to balance out the equations that makes the speed of light a constant. Cool.

Two last notes about length contraction.

#1. The length contraction would be perceived by both the people in the moving object and the people not in the moving object. Each would consider themselves at rest and the other objects moving.

Length Contraction

On the top: Trains DEF seem short from the frame of ABC
On the bottom: Trains ABC seem short from the frame of DEF

#2. The length contraction only occurs in the direction of motion. So, in our last example, the height of the train would remain the same even while the length was apparently decreasing.

That’s a pretty good primer on length contraction. I’ll do one more post on relative mass and then we’ll take a little break from physics to look at some other cool stuff.

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Framing the debate: relativity

15 Mar

Note: Albert Einstein’s birthday was yesterday, March 14th. I didn’t realize that until after I had decided to write a few facts about relativity. Sometimes you just get lucky.

I’ve seen a couple of links to a Popular Science article, “Warp Factor,” about a NASA engineer’s effort to develop an engine that could travel faster than the speed of light. I read it last night and I was rather skeptical, purely from a theoretical standpoint. Still, it did get me thinking about relativity and some of the funky things that happen at high speeds. Yet in order to talk about the weird stuff, you have to know what relativity means. Relative to what?

The most common way of explaining this, used by Daniel Frost Comstock in 1910 and Albert Einstein in 1917, is more or less this; imagine there is a man on a train moving past a man on the side of the tracks. Then, at the exact moment that the passenger and the bystander pass one another, a flash of light happens at the center of the train car. What does each man observe?

735px-Traincar_Relativity1.svgThe observer on the train experiences precisely what we’d expect him to see. He is traveling the same speed as the train, so relative to him the train is at rest. The flash of light happened at the center of the train, so the light reaches both ends of the train at the same time.

800px-Traincar_Relativity2.svg

The observer on the ground has a very different experience. The flash of light happened at the center of the train but relative to him the train was moving to the right. The left side of the train moved towards the light while the right side of the train moved away from the light. As a result, the light reaches the left side before it reaches the right side.

This raises a problem. One event happened, but there were two entirely different experiences. Which one is valid? The surprising answer is both! The key words in each of the explanations was the term relative to him and his frame of reference (hence the field of relativity).  Either of these can happen, depending on the observer’s frame of reference.

And that’s just the beginning of the weird stuff that happens in relativity (Still if you have any questions, now is a good time to ask). Stay tuned for the next episode where I’ll tell you all about time dilation.