# B What is the meaning of the speed of light "squared"?

1. May 5, 2016

### RandyD123

I'm not sure how the speed of light squared makes any sense? What is MASS X C2?

2. May 5, 2016

### Isaac0427

What about it doesn't make sense?
Energy.

If you are asking why is it squared, the answer lies in dimensional analysis.

3. May 5, 2016

### RandyD123

I see it as something like 22=4 and 42=16. So I don't get how we "double" the speed of light?

4. May 5, 2016

### Isaac0427

I'd review your exponents before asking these question. Squaring and doubling are not the same thing. 42=16, not 8.

5. May 5, 2016

I see 16?

6. May 5, 2016

### Staff: Mentor

Consider the units of mc2:

[kg]·[m/s]2 = kg·m2/s2 = J (joules) which is a unit of energy.

The c2 is basically a unit-conversion factor between kg and J.

7. May 5, 2016

### Isaac0427

Yes, 42=4*4=16. Similarly, c2=c*c.

8. May 5, 2016

### Isaac0427

I'd just like to expand on this so the OP doesn't get confused:
1 joule does NOT equal about 9*109 (c2) kilograms. It is saying that one kilogram of mass times c2 will give you one "kilogram" of energy (or I should say the amount of mass energy one kilogram of mass has).

9. May 6, 2016

### Gigel

Well, that's where the problem is, they are similar but not the same thing.

Take number 4. Then 4*4=16, which is a number 4 times higher.

Take c, which is about 3*108 m/s. Then c2=c*c=9*1016 m2/s2. Then c2 is not simply 3*108 times higher than c. It is 3*108 m/s higher than c. Their different physical dimensions (or units) mean that c and c2 are not the same kind of quantity, like both were numbers. They are different kinds of physical quantities and they cannot be compared.

It's just like one cannot say that 16 geese is 4 times more than 4 houses.

10. May 6, 2016

### the_emi_guy

Or a more precise analogy:
One cannot say that 16 square meters of area is 4 time more than 4 meters distance.

11. May 6, 2016

### Isaac0427

Correct, I chose to leave the units out of it however, just to explain the concept of exponentiation. I mentioned dimensional analysis in an earlier post.

12. May 6, 2016

### PeroK

Would it be wrong to say: "I have four times as many geese as chickens"?

The relationship between numbers and physical quantities is, perhaps, surprisingly subtle.

For example: if you have 4 rows of 4 marbles, then you have 16 marbles, not 16 square marbles. Yet, 4m by 4m is 16$m^2$.

Also, mass x velocity = momentum, but mass + velocity makes no sense.

Yet: 1 goose + 1 chicken makes sense, whereas, what would be meant by goose x chicken is not so clear!

13. May 6, 2016

### Gigel

Nope. In your example you are counting animals. If you were measuring your livestock, it would be different. That's why in estimating livestock cows and sheep don't count the same.

14. May 6, 2016

### PeroK

You're missing my point, though. We all know when we see a specific example what makes sense and what doesn't, but the general rules seem to be more complicated than at first sight.

For example, why can you multiply mass by velocity, but not colour by velocity? It's obvious you can't but why? That's not so obvious, when you think about it.

And, I think 4 x 4m = 16m (line) and 4m x 4m = 16$m^2$ (area) is quite interesting as well. It's like two different types of physical multiplication. I think that's quite interesting. Maybe it's just me.

15. May 6, 2016

### sophiecentaur

The operation of multiplication is 'outside' the choice of the dimensions of the quantities being multiplied.

16. May 6, 2016

### FactChecker

17. May 6, 2016

### ebos

A metre is one-dimensional. Squaring it makes it 2-dimensional. It is no longer a metre but a square metre. If you have 4 rows of 4 square metres you also have 16 square metres just like you have 16 marbles. Study dimensional analysis as mentioned above and all will become clear.

18. May 6, 2016

### Staff: Mentor

I seem to recall an ancient Greek theorem about squaring a circular cow, but for the life of me I can't find it...

It's simply a conversion factor to get from one unit of measurement to another. Kind of like how lbs = 0.453592*kg. I say "kind of" because I'm sure there are some subtle differences that I'm not familiar with.

19. May 7, 2016

### Mister T

When you double a speed you get another speed. When you square a speed you don't get another speed.

It makes sense to say something moves at speed c. It doesn't make sense to say something moves at speed c².

In the relation $E_o=mc^2$, $c^2$ is not a speed. It's simply a factor used to convert units of mass into units of energy. Note that there are systems of units where energy and mass already have the same units, in which case $c^2=1$, so $E_o=m$.

Regardless of the system of units used, $E_o=mc^2$ is a statement that mass $m$ and rest energy $E_o$ are equivalent. In other words, two things that we used to think were different are instead the same. That's all there is to it.

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