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## Main Question or Discussion Point

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

^{2}?- B
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I'm not sure how the speed of light squared makes any sense? What is MASS X C^{2}?

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Isaac0427

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What about it doesn't make sense?I'm not sure how the speed of light squared makes any sense?

Energy.What is MASS X C^{2}?

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

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- #4

Isaac0427

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I'd review your exponents before asking these question. Squaring and doubling are not the same thing. 4^{2}=4 and 4^{2}=16. So I don't get how we "double" the speed of light?

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I see 16?I'd review your exponents before asking these question. Squaring and doubling are not the same thing. 4^{2}=16, not 8.

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jtbell

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[kg]·[m/s]

The c

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Isaac0427

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- #8

Isaac0427

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I'd just like to expand on this so the OP doesn't get confused:^{2}:

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

The c^{2}is basically a unit-conversion factor between kg and J.

1 joule does NOT equal about 9*10

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Well, that's where the problem is, they are similar but not the same thing.Yes, 4^{2}=4*4=16. Similarly, c^{2}=c*c.

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

Take c, which is about 3*10

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

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Or a more precise analogy:It's just like one cannot say that 16 geese is 4 times more than 4 houses.

One cannot say that 16 square meters of area is 4 time more than 4 meters distance.

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Isaac0427

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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.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*10^{8}m/s. Then c^{2}=c*c=9*10^{16}m. Then c^{2}/s^{2}^{2}is not simply 3*10^{8}times higher than c. It is 3*10^{8}m/s higher than c. Their different physical dimensions (or units) mean that c and c^{2}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.

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Would it be wrong to say: "I have four times as many geese as chickens"?It's just like one cannot say that 16 geese is 4 times more than 4 houses.

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!

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

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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.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.

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.

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sophiecentaur

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The operation of multiplication is 'outside' the choice of the dimensions of the quantities being multiplied.It's like two different types of physical multiplication. I think that's quite interesting. Maybe it's just me.

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FactChecker

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(from https://en.wikipedia.org/wiki/Kinetic_energy#Kinetic_energy_of_rigid_bodies "By dropping weights from different heights into a block of clay, Willem 's Gravesande determined that their penetration depth was proportional to the square of their impact speed.").

That is probably where you should start if you are uncomfortable with c

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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.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!

- #18

Drakkith

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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.I'm not sure how the speed of light squared makes any sense? What is MASS X C^{2}?

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Mister T

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When you double a speed you get another speed. When you square a speed you don't get another speed.^{2}=4 and 4^{2}=16. So I don't get how we "double" the speed of light?

It makes sense to say something moves at speed

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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