AP Physics 1 1.3 Representing Motion

(a)

Students conduct an experiment to determine the acceleration a of a cart. The cart is released from rest at the top of the ramp at time t=0 and moves down the ramp. The x-axis is defined to be parallel to the ramp with its origin at the top, as shown in the figure. The students collect the data shown in the following table.

Table for Question (a) — AP Physics 1: Algebra-Based

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(i)

Indicate which quantities could be graphed to yield a straight line whose slope could be used to determine the acceleration a of the cart. You may use the remaining columns in the table, as needed, to record any quantities (including units) that are not already in the table.

Vertical axis: Horizontal axis:

[ 1 ]

(i) For indicating two quantities that, when graphed together, produce a straight line whose 1 point slope can be used to determine the acceleration a
Example Response
Vertical Axis: Position Horizontal Axis: Time squared

Position x
(m)

Time t
(s)

Time squared $t^{2} \left(\mathrm{s}^{2}\right)$
Time squared $t^{2}$ $\left(\mathrm{s}^{2}\right)$

0.06

0.39

0.15

0.14

0.59

0.35

0.24

0.77

0.59

0.37

0.96

0.92

0.55

1.20

1.44

(ii)

On the following grid, plot the appropriate quantities to create a graph that can be used to determine the acceleration a of the cart as it rolls down the ramp. Clearly scale and label all axes (including units), as appropriate. Draw a straight line that best represents the data.

Figure for Question (ii) — AP Physics 1: Algebra-Based

[ 3 ]

(ii) The axes have a linear scale and are identified (labels OR units) so that when graphed 1 point correctly, the data will span more than half of the horizontal and vertical axes
For plotting at least 4 of the data points correctly 1 point
For drawing a best-fit line that approximates the trend of the data 1 point

Example Response

Alternate Example Response

Scoring Note: The following tables represent the most common linearized graphs with the data that were used to determine the acceleration.

Graph: v vs. t

$v\left(\frac{\mathrm{~m}}{\mathrm{~s}}\right)$ $t(\mathrm{~s})$

0.15

0.20

0.40

0.49

0.56

0.68

0.68

0.87

0.75

1.08

Graph: 2 x vs. $t^2$ $2 x(\mathrm{~m})$ $t^{2}\left(\mathrm{~s}^{2}\right)$

0.12

0.15

0.28

0.35

0.48

0.59

0.74

0.92

1.10

1.44

Graph: Graph: $2v_{\text{avg}}$ vs. t $2 v_{\text {avg }}\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$ $t(\mathrm{~s})$

0.31

0.39

0.47

0.59

0.62

0.77

0.77

0.96

0.92

1.20

Graph: x vs. $\frac{1}{2}t^2$ }

$x(\mathrm{~m})$ $\frac{1}{2} t^{2}\left(\mathrm{~s}^{2}\right)$

0.06

0.08

0.14

0.17

0.24

0.30

0.37

0.46

0.55

0.72

Graph: Graph: $v_{\text{avg}}^2$ vs. x $v_{\text {avg }} 2\left(\frac{\mathrm{~m}^{2}}{\mathrm{~s}^{2}}\right)$ $x(\mathrm{~m})$

0.02

0.06

0.06

0.14

0.10

0.24

0.15

0.37

0.21

0.55

Graph: Graph: $\sqrt{x}$ vs. t $\sqrt{x}(\sqrt{\mathrm{~m}})$ $t(\mathrm{~s})$

0.24

0.39

0.37

0.59

0.49

0.77

0.61

0.96

0.74

1.20

(iii)

Using the line you drew in part (a)(ii), calculate an experimental value for the acceleration a of the cart as it rolls down the ramp.

[ 2 ]

(iii)
For attempting to find the slope, ( $\frac{\text { rise }}{\text { run }}$ ) or ( $\frac{\Delta y}{\Delta x}$ ), of the best-fit line drawn in part (a)(ii) 1 point
Scoring Note: An indication that a calculator was used for linear regression to determine the value of the slope may earn this point.
For using the slope in a valid kinematic equation to calculate the acceleration 1 point
Scoring Note: This point can be earned if evidence of a kinematic equation exists in graphed quantities (e.g., a graph of position as a function of $\frac{1}{2} t^{2}$ ).
Example Response

\begin{aligned} & \text { slope }=\frac{\Delta y}{\Delta x}=\frac{\Delta \text { position }}{\Delta \text { time }^{2}}=\frac{0.48 \mathrm{~m}-0.18 \mathrm{~m}}{1.2 \mathrm{~s}^{2}-0.4 \mathrm{~s}^{2}}=0.375 \frac{\mathrm{~m}}{\mathrm{~s}^{2}} \\ & \Delta x=v_{0} t+\frac{1}{2} a t^{2} \\ & \frac{\Delta x}{t^{2}}=\frac{1}{2} a \\ & \text { slope } \times 2=a \\ & a=0.75 \frac{\mathrm{~m}}{\mathrm{~s}^{2}} \end{aligned}

Total for part (a) 6 points

(b)

The students are asked to determine an experimental value for the acceleration due to gravity $g_{\text {exp }}$ using their data.

[ 2 ]

(i)

What additional quantities do the students need to measure in order to calculate $g_{\text {exp }}$ from a ?

[ 1 ]

(i) For indicating a quantity to be measured 1 point
Accept one of the following:

- The angle $\theta$ with the horizontal

- The height h and length L of the ramp

Scoring Note: Stating only the height needs to be measured can earn this point if an energy approach is used.

(ii)

Write an expression for the value of $g_{\text {exp }}$ in terms of a.

[ 1 ]

(ii) For providing a correct expression relating the acceleration of gravity to the acceleration 1 point measured
Scoring Note: If $\cos \theta$ is used, the response must specify that $\theta$ was measured from the vertical.
Example Response

\begin{aligned} & m g_{\exp } \sin \theta=m a \\ & g_{\exp }=\frac{a}{\sin \theta} \end{aligned}

OR

\begin{aligned} & \sin \theta=\frac{h}{L} \\ & g_{\exp }=\left(\frac{L}{h}\right) a \end{aligned}

OR

\begin{aligned} & m g_{\exp } h=\frac{1}{2} m v^{2} \\ & g_{\exp } h=\frac{1}{2} v^{2} \\ & v=\sqrt{2 g_{\exp } h} \\ & v=a t \\ & a t=\sqrt{2 g_{\exp } h} \\ & g_{\exp }=\frac{a^{2} t^{2}}{2 h} \end{aligned}

(c)

On the following graphs, sketch the position x and velocity v as functions of time t that correspond to the scenario shown while the cart moves up the ramp.