Question

show all your work for each part of the question. the parts within the question may not have equal weight.

students are given a small sphere, a stopwatch, and a meterstick and are asked to take measurements to create a graph that could be used to determine the acceleration due to gravity.

(a) describe a procedure for collecting data that can be graphed to determine the acceleration due to gravity. students have access only to the sphere, stopwatch, and meterstick. include any steps necessary to reduce experimental uncertainty.

note on your ap exam, you will handwrite your responses to free - response questions in a test booklet

(b) describe how the data collected in part (a) would be graphed, and how that graph would be analyzed to determine the acceleration due to gravity.

another group of students is given a cart on an inclined track. at the end of the track is a motion sensor which records the velocity of the cart as a function of time. the following table shows the data collected by the students.

Explanation:

Part (a)

1. Set up the experiment: Use the meterstick to measure a height \( h \) from which the sphere will be dropped. Ensure the height is measured accurately from the release point to the ground (or a target surface).
2. Release the sphere: Drop the sphere from rest from the measured height \( h \). Use the stopwatch to measure the time \( t \) it takes for the sphere to fall the distance \( h \).
3. Repeat for multiple heights: Repeat the process for several different heights (e.g., 0.2 m, 0.4 m, 0.6 m, 0.8 m, 1.0 m). For each height, perform multiple trials (e.g., 3 - 5 trials) and take the average time for each height to reduce random errors.
4. Record data: For each height \( h \), record the average time \( t \) taken for the sphere to fall.

Part (b)

1. Graph setup: The kinematic equation for free - fall (assuming initial velocity \( v_0 = 0\)) is \( h=\frac{1}{2}gt^{2}\), which is in the form of a quadratic equation \( y = ax^{2}\) where \( y = h\), \( x = t\), and \( a=\frac{g}{2}\). So, we should graph \( h \) (vertical axis, dependent variable) versus \( t^{2}\) (horizontal axis, independent variable).
2. Graph analysis: If we plot \( h \) against \( t^{2}\), the graph should be a straight line (since \( h\) is proportional to \( t^{2}\) for free - fall with \( v_0 = 0\)). The slope \( m\) of the line \( h=\frac{1}{2}gt^{2}\) is related to \( g\) by the equation \( m=\frac{g}{2}\). So, to find \( g\), we calculate the slope of the best - fit line through the data points on the \( h\) vs. \( t^{2}\) graph and then use the formula \( g = 2m\).

Part (a) Answer:

1. Measure a height \( h \) with the meterstick.
2. Drop the sphere from rest from height \( h \), measure fall time \( t \) with the stopwatch.
3. Repeat for multiple heights, take multiple trials per height for average time.
4. Record \( h \) and average \( t \) for each height.

Part (b) Answer:

1. Graph \( h \) (y - axis) vs. \( t^{2}\) (x - axis).
2. The graph is a straight line. Calculate the slope \( m \) of the line. Then \( g = 2m \) (from \( h=\frac{1}{2}gt^{2}\), so slope \( m=\frac{g}{2}\)).

Answer:

1. Graph setup: The kinematic equation for free - fall (assuming initial velocity \( v_0 = 0\)) is \( h=\frac{1}{2}gt^{2}\), which is in the form of a quadratic equation \( y = ax^{2}\) where \( y = h\), \( x = t\), and \( a=\frac{g}{2}\). So, we should graph \( h \) (vertical axis, dependent variable) versus \( t^{2}\) (horizontal axis, independent variable).
2. Graph analysis: If we plot \( h \) against \( t^{2}\), the graph should be a straight line (since \( h\) is proportional to \( t^{2}\) for free - fall with \( v_0 = 0\)). The slope \( m\) of the line \( h=\frac{1}{2}gt^{2}\) is related to \( g\) by the equation \( m=\frac{g}{2}\). So, to find \( g\), we calculate the slope of the best - fit line through the data points on the \( h\) vs. \( t^{2}\) graph and then use the formula \( g = 2m\).

Part (a) Answer:

1. Measure a height \( h \) with the meterstick.
2. Drop the sphere from rest from height \( h \), measure fall time \( t \) with the stopwatch.
3. Repeat for multiple heights, take multiple trials per height for average time.
4. Record \( h \) and average \( t \) for each height.

Part (b) Answer:

1. Graph \( h \) (y - axis) vs. \( t^{2}\) (x - axis).
2. The graph is a straight line. Calculate the slope \( m \) of the line. Then \( g = 2m \) (from \( h=\frac{1}{2}gt^{2}\), so slope \( m=\frac{g}{2}\)).