Special relativity is perhaps one of the most interesting parts of the HSC Physics Course. It is also, however, one of the most difficult to wrap your head around. Understandably so; you are attempting to describe and explain one of the most groundbreaking theories in the history of Classical Physics, one coined by Einstein himself. Did you expect it to be easy?
Now, the HSC course doesn’t expect you to know absolutely everything about this, but you do need a very solid understanding, because
special relativity is worth at least a handful of marks in every HSC paper. To not understand it is to throw away those marks. So, by the end of this resource, I aim to make you an expert.
As a quick side note, this resource does assume a decent understanding of relative velocities. A quick example: You are in a car travelling 100, and you throw a tennis ball out the back window. You can throw a ball at 20. You see the ball travel backwards at 20, but someone watching from the side of the road would still see the ball travelling forward, at 80. Why? You were travelling 100 to begin with, so really, you just subtracted 20 from the balls speed. If you are struggling with this concept, create a thread or jump on the Q+A forum and I’ll help you out! The Theory of Special Relativity was “created” by Einstein 1905. Essentially, it extrapolates on Galilean Relativity by examining the properties of objects and their motion at relativistic speeds. In English: It takes what they already knew, and expands it, to make it work with objects travelling close to the speed of light. It contains two main principles/assumptions:
- All inertial frames of reference are equally valid
- The speed of light is constant in all reference frames, and this speed cannot be exceeded
This was a crazy thought in 1905, when it was always believed that light would change speed based on which way it was travelling, and how you observed it. Michelson and Morley, a pretty clever scientific duo, even tried to detect the changes in speed. You learn about the experiment in this topic as well, and the null result they obtained played perfectly into Einstein’s theory. Now, these assumptions create some pretty interesting scenarios.
As an example let’s consider the infamous train experiment. You are on a train travelling at the speed of light, with no windows, no bumps in the rail. It feels identical to standing still in your room; but you are moving fast. Wicked fast. You are in an inertial frame of reference; travelling at a constant velocity with no acceleration.
Now, you hold up a mirror. Can you see yourself?
Yes, all inertial frames of reference are equally valid, you can see yourself just the same as at home! No, the speed of light is constant in all reference frames. If you are already travelling at the speed of light, the light from the mirror will never catch up to you! Wait… What? How can both of Einstein’s assumptions be true? If all inertial frames of reference are equal, I shouldn’t be able to tell I am moving at all, but how does the light catch up to me if I’m moving just as fast away from it. There doesn’t seem to be a solution… But there is.
Time slows down. Think about it. To see yourself, the light has to travel from you, to the mirror, back to you. Standing still, that distance is . On a train moving very close to the speed of light, that distance becomes much larger, as the light has to catch up to you. Let’s call that distance (capital). If the speed of light is a constant , we can then write:
If D is bigger than d, and c is constant, then the only way that this works, is if the t1 and t2 are different. Specifically, t2 is bigger. It’s kind of like an equivalent fractions sort of thing.
This is a mathematical description of a phenomenon known as time dilation. Time actually passes by more slowly at higher velocities. At speeds we are used to, we don’t notice, but at relativistic speeds the difference is quite shocking.
Some more results:
- Distances actually become shorter at higher speeds. This is actually kind of intuitive, at least in my head, that when you zoom faster and faster, the stuff you are looking at gets compressed. This phenomenon is called Length Contraction
- The mass of an object increases at higher speeds. To imagine this, picture an object travelling at close to the speed of light, say, less. We exert a force enough to cause an acceleration of on the object. But the object can’t exceed the speed of light, what gives? Well, let’s think about . If acceleration (and thus, speed) isn’t increasing then there is only one thing left… Mass. Yep, objects get heavier the faster you go. This is called Mass Dilation.
- Finally, in the scenario above, we have imparted energy to the object. Its speed doesn’t go up; neither does its temperature or anything else. The only change is that the object gets heavier. You guessed it, mass and energy are the same thing. This is called, funnily enough, the equivalence of mass and energy.
Right, so that is the theory. If you know it, you can handle any written response. But what about the math?
Thankfully, most questions in this area are substitution questions, with the following formulae.
For time dilation:
For mass dilation:
For length contraction:
These formulas all look fairly similar. The letters with v-subscript (Tv, Cv, Lv) are the relativistic values, which are experienced by the moving object. The letters with o-subscript are the values as seen by a stationary observer, the actual measurements without the influence of Special Relativity. Often, the most confusing part is getting these two the right way around! The speed of the object is v, and the speed of light is c.
The formula for mass/energy equivalence should be easy to remember, it’s only the most famous formula of all time:
Put that into perspective: 1 kilogram of mass is equivalent to 9 thousand million million joules of energy, or, pretty close to the total energy from the sun that strikes the earth per second. If we could convert mass to energy with anywhere near 100% efficiency, then our energy woes would be over. Crazy!
Let’s look at a couple of examples.
Example 1: A new home-based energy generation method converts household waste to electrical energy with 30% efficiency. How much energy could a household with 2 kilograms of waste a day generate, per day?
This one is fairly straightforward if you break it down. First, calculate the energy in 2 kilograms of waste.
Then, we just get the final value by taking 30% of this value:
Example 2: An astronaut leaves his wife in the year 2016, and returns in 2021. The astronaut travels at an average speed of half the speed of light throughout the journey. How long will the astronaut think he is in space?
What this question is really asking is, what happens to time while in the frame of reference moving at half the speed of light.
We want to know the relativistic value, and we are given the typical value (6 years). Beyond the conversion of years to seconds (SI Units), this is a very typical question.
Now we use the formula:
Or about 1.15 years. The astronaut experiences time more slowly, so a year to them is about 5.5 years to us at these sorts of speeds!
Hopefully this explanation of Special Relativity, with a few examples, is helpful in the lead up to your half yearly. It is a very tricky concept, so if you are having trouble, post below and I’ll be happy to lend a hand!
To ask a question, just make an ATAR Notes account here.
and post it below! I'd be happy to help out in any way I can 
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