Potential Energy of the system

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Homework Statement


In an experiment, a student wishes to use a spring to accelerate a cart along a horizontal level track. The spring is attached to the left end of the track and produces a non-linear force of magnitude Fs = Ass + Bs, where s is the distance the spring is compressed in meters. A measuring tape, marked in centimeters, is attached to the side of the track.

Derive an expression for the potential energy ∪ as a function of the compression s. Express your answer in terms of A, B, s, and fundamental constants as appropriate.

Homework Equations


Us = 1/2 * k * (Δx)s (changed x to s in this problem)
Fs = -k * Δx (changed x to s in this problem)
ΔE = ∫F * dr

The Attempt at a Solution


So I know that we need to find an expression for potential energy and since they said to included: A, B, s. I wanted a way to include Fs into potential energy. So I set the formula for Fs equal to their version of Fs (so, (-k* Δx) = ( Ass + Bs)) and got k = (Ass + Bs) / Δs. And using this k, I plugged it into Us and got ((Ass + Bs) * Δs) / 2.

However, my teacher showed us that the way you were supposed to do it was using the formula for ΔE using the integral. But why would the ΔE be the same as ∪s? The final answer was (1/3 * As3 ) + (1/3 * Bs2).

Thanks so much for the help! (I think this belonged more to the introductory physics homework forum so I posted there too and tried to delete this but I don't think I can. But help would be appreciated if available, but if this doesn't belong here I understand for my mistake.)
 
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