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So I had my hand held for the derivations for the transformations of length and time in special relativity, and I wanted to see if I could do the same thing with mass. I came up with something, but I'm not sure if my methodology is correct.

I said that Energy is the force applied on something for a distance.

E = F*d

If you were "standing still" and saw a spaceship accelerating with thrust F for distance d, you would say that the energy exerted was F*d

However, if the person in the spaceship were to say how much energy was exerted, then the would also multiply a distance by a force. However, they would be traveling at a velocity, v, and from their point of view, they were not being accelerated for a distance, d, but rater a distance of

d*sqrt{1-v

So they would say that E = F'*d*sqrt{1-v

The total sum of energy in a system shouldn't change for people in different reference frames. So

F'*d*sqrt{1-v

m'*a*sqrt{1-v

And because the rate of acceleration can be known to anybody in any reference frame,

m' = m/sqrt{1-v

And that's how I did it

I'm a little wary of this little derivation for a few reasons, but I get the correct result. So I'm curious if this works because my methodology is right, or because I made multiple mistakes that "canceled each other out."

Thanks

I said that Energy is the force applied on something for a distance.

E = F*d

If you were "standing still" and saw a spaceship accelerating with thrust F for distance d, you would say that the energy exerted was F*d

However, if the person in the spaceship were to say how much energy was exerted, then the would also multiply a distance by a force. However, they would be traveling at a velocity, v, and from their point of view, they were not being accelerated for a distance, d, but rater a distance of

d*sqrt{1-v

^{2}/c^{2}}So they would say that E = F'*d*sqrt{1-v

^{2}/c^{2}}The total sum of energy in a system shouldn't change for people in different reference frames. So

F'*d*sqrt{1-v

^{2}/c^{2}} = F*dm'*a*sqrt{1-v

^{2}/c^{2}} = m*aAnd because the rate of acceleration can be known to anybody in any reference frame,

m' = m/sqrt{1-v

^{2}/c^{2}}And that's how I did it

I'm a little wary of this little derivation for a few reasons, but I get the correct result. So I'm curious if this works because my methodology is right, or because I made multiple mistakes that "canceled each other out."

Thanks

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