# Einstein's Special Theory of Relativity: Analysis & Practice Problems

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• 0:03 Special Theory of Relativity
• 1:18 Equations Summary
• 3:19 Practice Problems
• 5:27 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After completing this lesson, you should be able to explain what Einstein's special theory of relativity is and use equations from the theory to solve problems. A short quiz will follow.

## What Is the Special Theory of Relativity?

Special relativity is a physics theory proposed by Albert Einstein in the early 20th century. Its implications are many and complex, but there are a few basic ideas on which it is based. Special relativity says that:

• There is no absolute frame of reference
• The laws of physics hold in every frame of reference
• The speed of light is a universal constant that is always the same

Let's imagine that you're on a bus, and that bus is traveling east at 30 meters per second. But then, suddenly, you run towards the back of the bus - you run west at a velocity of 5 meters per second relative to the floor of the bus. What then, is your velocity? Is it 5 meters per second because that's how fast you're running? Is it still 30 meters per second because you're inside the bus? Or, is it 25 meters per second (30-5) because that's how fast your position relative to the earth is changing?

According to special relativity, all these answers are correct. There is no absolute frame of reference. In fact, right now you're shooting around the sun, and the solar system is orbiting around the center of the galaxy, and the whole galaxy is also moving. So, how fast you're moving really does depend on your frame of reference.

## Equations Summary

The concepts of special relativity might seem extraordinarily basic, but the consequences are far reaching when you analyze them mathematically. There are three main consequences we can use in solving problems: relativistic mass, length contraction and time dilation.

Relativistic mass is the idea that as you increase speed, your effective mass also increases because it becomes increasingly hard to accelerate you. This effect is only noticeable at speeds that are a large proportion of the speed of light. You can't tell in everyday life. Relativistic mass can be calculated by this equation.

Length contraction is the idea that as you increase speed, the length of an object in the direction of motion contracts (or shortens). And that's represented by this equation.

And last of all, time dilation is the idea that as you increase speed, the rate at which time passes decreases relative to an observer. On a spaceship moving close to the speed of light, the clock might seem to be ticking at the right rate, but when you return to Earth you'll find that much more time has passed there - you'll have traveled into the future! Time dilation is represented by this equation.

In these three equations, a zero represents what you experience in the moving frame of reference - meaning, on the spaceship. So T0 is the time you measure in the moving frame of reference. L0 is the length of an object that you measure in the moving frame of reference. And m0 is the mass of an object in the moving frame of reference, or in other words, the mass that it would have if it wasn't moving. These values without zeros are the values you measure from the point of view of someone observing from a stationary frame of reference (a place where v=0). V is the velocity of the moving frame of reference... the velocity of the space ship. And c is the speed of light, which is always 3 x 10^8 meters per second.

## Practice Problems

Practice Problem 1:

A spaceship is moving at 0.3 times the speed of light relative to the earth. If the spaceship has a length of 15 meters, how long will it appear when observed from the earth?

First of all, we should write down what we know. The velocity, v, of the moving frame of reference is 0.3 times the speed of light. We could actually take the speed of light and multiply it by 0.3, but let's just leave this as 0.3c for now. You'll see why later.

The length of the ship from the point of view of the people traveling in it is 15 meters. So L0 equals 15 meters.

Plug these into our length contraction equation,

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