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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