Four people (A-D) push the same heavy ball 100 m along a track. According to the graph (will draw inside!) who used the least force? person A, person B, person C, or person D?

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Four people (A-D) push the same heavy ball 100 m along a track. According to the graph (will draw inside!) who used the least force? person A, person B, person C, or person D?

Physics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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|dw:1433089983896:dw|
x axis is "Velocity (m/s)" and y-axis is "Power (W)" too messy trying to write! haha :P
I'm pondering...

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Other answers:

:)
is the motion of the ball at constant speed?
yes i believe so :/ it does not indicate but it says same heavy ball along the track so i am assuming so?
I think so
ok! so what do you think for this problem? I am not sure who would have used the least force :/
please wait...
ok!
I think that there are many variables which can determine the motion of the ball, nevertheless, the minimum force is associated to the minimum power
ohh so that would be person D?
yes!
yay!! thank you!!
please wait,I'm pondering again...
ooh okie!
wait so it is not person D?
here is my reasoning: space d traveled by the ball is given by the subsequent formula: \[d = \frac{1}{2}a{t^2}\] since the ball perform an accelerate motion, right?
yes:) so who would it be then?
so time t needed to travel that space d is: \[t = \sqrt {\frac{{2d}}{a}} \] a is the acceleration
now we consider two forces, namely F_1 which produces an acceleration a_1, and F_2 which produces an acceleration a_2
so we can write, since d is the same in both cases: \[{t_1} = \sqrt {\frac{{2d}}{{{a_1}}}} ,\quad {t_2} = \sqrt {\frac{{2d}}{{{a_2}}}} \]
ok!
i am confused so it is still person D?
now, the powers of each forces, are given by the change of kinetic energy over time, so we can write: \[{W_1} = \frac{1}{2}\frac{{mv_1^2}}{{{t_1}}},\quad {W_2} = \frac{1}{2}\frac{{mv_2^2}}{{{t_2}}}\]
ok!
now we can simplify those formulas as follows: \[{W_1} = \frac{{F_1^{3/2}{d^{1/2}}}}{{\sqrt 2 m}},\quad {W_2} = \frac{{F_2^{3/2}{d^{1/2}}}}{{\sqrt 2 m}}\]
so, if we consider their ratio, we get: \[\Large \frac{{{W_1}}}{{{W_2}}} = {\left( {\frac{{{F_1}}}{{{F_2}}}} \right)^{3/2}}\] now if F_1 is less than F_2, then also W_1 is less than W_2, so our statement is proved!
yay! so person D it is! thank you:D
thank you!! :)

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