Double the speed

Homework Statement

A proton is accelerated from rest through a potential difference of 500MV. Classical mechanics indicates that quadrupling the potential difference would double the speed. Were a classical analysis valid, what speed would result from a 2000MV potential difference?

4V->2v

The Attempt at a Solution

We were first asked to find the original speed of the proton going through 500MV which is 0.76c Since 2000MV = 4*500MV and 4V means v is doubled, why can't I just double 0.76c to get the new speed for the proton accelerated through the 2000MV?

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SammyS
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Homework Statement

A proton is accelerated from rest through a potential difference of 500MV. Classical mechanics indicates that quadrupling the potential difference would double the speed. Were a classical analysis valid, what speed would result from a 2000MV potential difference?

4V->2v

The Attempt at a Solution

We were first asked to find the original speed of the proton going through 500MV which is 0.76c Since 2000MV = 4*500MV and 4V means v is doubled, why can't I just double 0.76c to get the new speed for the proton accelerated through the 2000MV?
I believe that you are asked to do precisely that.

Yet the answer is 2.07c :/

the book solves it by 4*(the original KE, 8*10^-11 J) = .5 * m * v^2 then solving for v.......but I really just don't see why I can't logically just straight-up double the velocity...