$1.$ Perform a linear fit to that portion of the v$-$t graph where the cart was moving freely. Print or sketch this v$-$t graph. Write the equation that represents the relationship between the velocity and time; be sure to record the value and units of the slope and the vertical intercept.

On this v$-$t graph identify:

$$ Where the cart was being pushed by your hand

$$ Where the cart was rolling freely up the ramp

$$ The velocity of the cart when it was farthest from its initial position

$$ Where the cart was rolling freely down the ramp

$2.$ The slope of a graph represents the rate of change of the variables that were plotted. What can you say about the rate of change of the velocity as a function of time while the cart was rolling freely? In your discussion, you will give a name to this quantity. What is the significance of the algebraic sign of the slope?

$3.$ Compare the value of your slope to those of others in the class. What relationship appears to exist between the value of the slope and the extent to which you elevate the track?

$4.$ The vertical intercept of the equation of the line you fit to the v$-$t graph represents what the velocity of the cart would have been at time t $=0$ had it been accelerating from the moment you began collecting data. Suggest a reasonable name for this quantity. Now write a general equation relating the velocity and time for an object moving with constant acceleration

$5.$ The position time graph of an object that is constantly accelerating should appear parabolic. Use the Curve Fit function of your data analysis program to fit a quadratic equation to that portion of the x$-$t graph where the cart was moving freely. Note the values of the A and B parameters in the quadratic equation. You will have to provide the units.

$6.$ Compare these parameters $($values and units$)$ to the slope and intercept of the line used to fit the v$-$t graph. Now write a general equation relating the position and time for an objectLogger Pro $-$ Motion on an incline$-$sensor top $(2)*$

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$\backslash$table$[[,$Latest$],[,\backslash$table$[[$Time$],[($s$)]],\backslash$table$[[$Position$],[($m$)]],\backslash$table$[[$Velocity$],[($m$/$s$)]]],[1,,,],[2,,,],[3,,,],[4,,,],[5,,,],[6,,,],[7,,,],[8,,,],[9,,,],[10,,,],[11,,,],[12,,,],[13,,,],[14,,,],[15,,,],[16,,,],[17,,,],[18,,,],[19,,,],[20,,,],[21,,,],[22,,,],[23,,,],[24,,,],[25,,,],[26,,,],[27,,,],[28,,,],[20,,,]]$

Position

$0\mathrm{.}000m$ undergoing constant acceleration.