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Lesson 3: Length Contraction

Lesson 1: Relativity - What is it?

  1. Fermilab's Time Dilation Challenge.
  2. The Basics of Relativity (6 seconds)
  3. The Relativity Game - Challenge what you know!

Note: For Fermilab's Time Dilation Challenge and The Relativity Game, you need Shockwave. You may painlessly Download Shockwave here if you do not have it.

Lesson 2: Time Dilation

  1. The equation.
  2. Where does that come from?
  3. I still don't get it! Give me the basketball analogy.
  4. So what? There's an equation. How do I use the equation in the game?
  5. Practice Problems.
  6. Examples to aid your practice.

Lesson 3: Length Contraction

  1. The equation.
  2. Where does that come from?
  3. I still don't get it! Give me the pole and the barn analogy.
  4. So what? There's an equation. How do I use the equation in the game?
  5. Practice Problems.
  6. Examples to aid your practice.

Lesson 4: Relativistic Mass

  1. The equation.
  2. Where does that come from?
  3. I still don't get it! Give me the space pool analogy.
  4. So what? There's an equation. How do I use the equation in the game?
  5. Practice Problems.
  6. Examples to aid your practice.

Where does that come from?

Lets consider a ship that the earth observe to be moving at 50% c. Both a moving frame and a stationary frame want to measure the length of a stick with respect to a vertical line in our frame. Both the earth's frame and the ship's frame will determine the length of the stick by knowing v, and measuring time within their own reference frame. Once v and t are known, each frame can determine L by using the equation for distance, d = v*t (review this equation).

Ship's Frame of Reference:

The ship measures the time for the stick to pass the reference line that is stationary in the earth's frame of reference. In this case, the time is being measured with respect to the earth's reference line. Therefore, the moving frame (mover for mover) is defined to be the earth's frame.

With the ship measuring its stationary frame to have the earth speeding by with a speed of 50 % c, it measures a time for the mover by the stationary frame to be 2.31 seconds. Because the rocket's passenger knows the equation for distance or length (review this equation) he measures the stick's length to be:

d = v*t

Lo = v*t Eqn. 1

Lo = (150,000,000 m/s)(2.31 s)

Lo = 346,400,000 meters.

The keen reader notices that we used Lo for length, which is the length of the mover by the mover and yet used t for time which is the time for the mover by the stationary frame. This is not a mistake. The meter stick being measured is at rest on the ship's frame, thus we used Lo. However, the time measurement is made with respect to the earth's frame so the earth uses to and the ship uses t!

The Earth's Frame of Reference:

Now consider the same time measurement taken from the earth's frame (the moving frame). The earth's frame measures the time to be different as we saw in time dilation.

With the earth measuring its frame to have the ship speeding by with a speed of 50 % c, it measures a time for the mover by the mover to be 2.00 seconds. Because the earth's passenger knows the equation for distance or length (review this equation) he measures the stick's length to be:

d = vt

L=vto. Eqn. 2

L = (150,000,000 m/s)(2.00 s)

Lo = 300,000,000 meters.

However, from above, we see that the distance covered for the mover by the mover is Lo and from:

Lo=vt. Eqn. 1

Now let's remember that:

Eqn. 3

So when we consider Equation 1 and Equation 3 together, we get:

Eqn. 4

Now, if we rearrange Equation 2 and substitute it into equation 4, we get:

Question:

Which is ALWAYS greater, L or Lo?

What is L, Lo, v, and c?

Lesson 3: Length Contraction

  1. The equation.
  2. Where does that come from?
  3. I still don't get it! Give me the Alpha Centuri example.
  4. So what? There's an equation. How do I use the equation in the game?
  5. Practice Problems.
  6. Examples to aid your practice.

 


© Brian Wegley, 1998
Comments and suggestions can be sent by e-mail to
Brian Wegley of Glenbrook South High School..
This page last updated on 7/23/98.