Graph relationships between speed & distance?

[arc5005]

Background question information:
A student applies a force to a stalled car over a distance delta x to increase its kinetic energy.

Please look at this image for the question & pictures of the graphs: IMG 2821

A. A
B. B
C. C
D. D




C) C

Their answer:
As the car is pushed farther and farther, it builds up kinetic energy. As the KE increases, its v2 increases. The relationship is v2 α x, which is not linear, so eliminate choices A & B.
It is best to plug in numbers and see how the two values relate to one another. If v is doubled, then x must increase by a factor of 4. This means that x changes more than v changes, so the graph should bend towards the x-axis. Remember,"graphs curve towards the axis of greatest change."

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QUestioN:
Why exactly is the answer C & not D???

W = F * d
W = deltaK = KE = 1/2mvf^2 - 1/2mvi^2

F * d = 1/2mvf^2

d = 1/2mvf2 / F

d is proportional to v^2

If you increase d by a factor of 2, then v^2 would increase by a factor of 4? right? or do I have this reversed?

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[Lawpy]

You have the right idea but note that if d is proportional to v^2, this means if v is increased by a factor of 2, d is increased by a factor of 2^2 or 4. Since the graphs place v in the vertical (or dependent) axis and d in the horizontal (or independent) axis, you want to express v in terms of d (or x as the problem states but the variable used doesn't matter). d being proportional to v^2 is the same thing as v being proportional to square root of d, hence C is the correct answer since it shows the square root curve.

 

[arc5005]

You have the right idea but note that if d is proportional to v^2, this means if v is increased by a factor of 2, d is increased by a factor of 2^2 or 4. Since the graphs place v in the vertical (or dependent) axis and d in the horizontal (or independent) axis, you want to express v in terms of d (or x as the problem states but the variable used doesn't matter). d being proportional to v^2 is the same thing as v being proportional to square root of d, hence C is the correct answer since it shows the square root curve.

okay thank you.