Students use a stretched elastic band to launch carts of known mass horizontally on a track. The elastic bands exert a force F, which is nonlinear as a function of the distance xx that the band is stretched. A meterstick is used to measure xx , and a motion detector is used to measure the resulting maximum speed of the carts. This procedure is repeated to gather data for several different values of x. Which of the following graphs, if linear, supports the hypothesis that F is a function of x^2? A. A graph of the cart's maximum speed as a function of x
B. A graph of the cart's maximum speed as a function of x^2
C. A graph of the cart's maximum speed squared as a function of x
D. A graph of the cart's maximum speed squared as a function of x^2
E. A graph of the cart's maximum speed squared as a function of x^3

the correct answer is  E

A graph of the cart's maximum speed squared as a function of x^3

Explanation:

For this exercise let's use Newton's second law

        F = m a

force has the form

        F = k x²

and acceleration is related to velocity

        a = dv / dt

Let's use the chain rule or L'Hospital

        a = dv /dx   dx/dt

        a = dv /dx    v

let's substitute

     k x² = m v dv / dx

     k /m x² dx = v dv

we integrate

     k /m    x³ /3 = v² / 2

     v² = (2k /3m)   x³

This is the expression for the variation of the speed as a function of the position, to make a linear graph realism the changes of variable

      y = v²

     x´ = x³

       y = (2k/3m)  x´

if we graph y vs x 'we have a linear graph whose slope is

      m = 2k / 3m

By reviewing the different answers, the correct answer is  E

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