# Vertical velocity and integration

## Homework Statement

a list of data is given for a seat on a ferris wheel that is traveling in a circle. the data is time vs its vertical velocity. the question is to estimate the diameter of the ferris wheel.

n/a

## The Attempt at a Solution

what i did was integrate the function from 0 to 30 using a reimann sum (which gave me just an approximation because there was no function given...just the vertical velocities every 5 seconds).

i chose o to 30 because at 30 seconds, the vertical velocity became 0 again. this implied to me that the ferris wheel would be at the top because the vertical velocity is 0 at the top.

so i integrated by adding up all the rectangles from 0 to 30 (each with equal base length) and got some value. and since im integrating a velocity function, i should come out with a displacement function for the seat on the ferris wheel.

now my question is, does this value represent the actual distance traveled of the ferris wheel (ie, how far the seat traveled along the circumference) OR does it represent the vertical displacement of the seat (ie, how far it traveled upwards)?

if its the first, then i can find the diameter by doubling that value to get the ful circumference, then setting that equal to (pi)d and getting my diameter.

if its the second, then my answer should automatically be the approximation of the diameter since the vertical displacement from the bottom to the top is the diameter of the ferris wheel.

which one is correct? am i logically justifying it correctly?

thanks.

SGT
You say that at 30 seconds the velocity is zero again. Does it mean that it started at zero?
If so, in 30 seconds the seat travelled from up to down to up again, so the vertical displacement is twice the diameter.

Well... I'm not sure I understand fully but I don't think this requires to jump to integrals... If the vertical velocity goes to 0 at 30 seconds interval, and the actual velocity around the circle is constant, what happens mid way through these 30 seconds? That's all you would need to think about in order to solve this...

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You say that at 30 seconds the velocity is zero again. Does it mean that it started at zero?
If so, in 30 seconds the seat travelled from up to down to up again, so the vertical displacement is twice the diameter.

it started at zero, then hit 0 again at 30 seconds and 60 seconds. which i believe means it was zero to begin with at the bottom, then 30 seconds later it hit the top (0 vertical velocity), then 60 seconds again at the bottom (where it started).

so, if my assumption is correct, and the integral of this velocity function is simply the vertical displacement only, then integrating from 0 to 30 is the diameter itself.

Well... I'm not sure I understand fully but I don't think this requires to jump to integrals... If the vertical velocity goes to 0 at 30 seconds interval, and the actual velocity around the circle is constant, what happens mid way through these 30 seconds? That's all you would need to think about in order to solve this...

yeah but how could i possibly get the diameter of the ferris wheel from that? many different wheels of all different diameters could all have the same period. the question asked for the diameter of the wheel (hence i tried using integrals).

SGT
it started at zero, then hit 0 again at 30 seconds and 60 seconds. which i believe means it was zero to begin with at the bottom, then 30 seconds later it hit the top (0 vertical velocity), then 60 seconds again at the bottom (where it started).

so, if my assumption is correct, and the integral of this velocity function is simply the vertical displacement only, then integrating from 0 to 30 is the diameter itself.

Integration from 0 to 30 is twice the diameter. It is the time for going from up to down to up again.

Integration from 0 to 30 is twice the diameter. It is the time for going from up to down to up again.

why is that? isnt the vertical velocity at the top also 0? therefore from 0 to 30 should be from bottom to top once - hence its the diameter.

but again thats using the assumption that integrating the vertical velocity gives you only the vertical displacement and not the actual distance traveled by the ferris wheel (which would correlate more with the circumference) - thats my main question.

HallsofIvy