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

given data:

determine the horse power (h.p) = ?

force applied to move the box(f) = 10 pounds,

box lifted from a distance         (d) = 3 feet

time taken to lift                         (t) = 10 sec,

the horse power is defined as amount of work required to move a load. the horse power is equal to 550 foot-pound per second

                  1 h.p   = 550 foot-pound/second

it can also be defined as the the load that need to be move a body for certain distance in a certain amount of time.

mathematically,

                      h.p = (force × distance)/ time

                            = (10 × 3)/ 10

                            = 30/10

                            = 3 foot -pound/sec

hence, the horse power is needed to move the box is 3 foot-pound/second

important note: 1 h.p = 550 foot-pound/second = 746 watts

mistakes are ok