Conversion of Energy problem

[Halcyon32]

Hi, I have some confusion regarding the following question:

The answer is B but I'm confused as to why it is not D. The answer key cites the fact that since h is proportional to v^2, doubling h would cause v to increase by a factor of sqrt2. I would think that since h is doubled, v^2 would also be doubled which translates into a proportionality of 4v. Or would it actually be 4v^2 which still ultimately boils down to technically being proportional to sqrt 2. Here is what the answer key specifically says:

Thanks in advance for all your help!

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[FMDoge2020]

2gh=v^2
So if h is doubled , it would be
2(2gh)=(answer x v)^2
(sqrt2)^2 gives you 2
Results in
2(2gh)=(sqrt(2)x v)^2=2(v^2)

If v is quadrupled, then (4v)^2would give you 16(v)^2 this means that 2gh would need to be multiply by 8:
8(2gh)=(4v)^2=16(v)^2

So you would need to raise the height 8times to achieve a quadrupled velocity


Sent from my SM-N920P using SDN mobile

 

[Halcyon32]

2gh=v^2
So if h is doubled , it would be
2(2gh)=(answer x v)^2
(sqrt2)^2 gives you 2
Results in
2(2gh)=(sqrt(2)x v)^2=2(v^2)

If v is quadrupled, then (4v)^2would give you 16(v)^2 this means that 2gh would need to be multiply by 8:
8(2gh)=(4v)^2=16(v)^2

So you would need to raise the height 8times to achieve a quadrupled velocity


Sent from my SM-N920P using SDN mobile

Shouldnt the first equation be 2gh=2(v^2)? Why are you doubling the potential energy portion but not the kinetic energy portion?

EDIT: nvm I forgot the 1/2 part of the kinetic energy portion. This makes sense. Thank you!