"Uniformly" Accelerated Motion

Introduction: Some experiments yield clearcut, unequivocal data; some experiments do not. However, with enough data, some results can be deemed as trustworthy. In this lab we will take lots of data from primitive equipment and analyze it in the spreadsheet.

Theory: The theory behind uniformly accelerated motion is simple. Let a falling weight drag a tape through an apparatus which marks it at regular time intervals, t. Given this set of coordinates, the distance between each pair can be calculated. If the time for each interval is known, the quotient of x/t yields the average velocity over the interval. (This also happens to be the velocity halfway between the coordinates, called the midmark velocity.) When one subtracts consecutive average (or midmark) velocities, and divides the differences by the same time interval, one arrives at a set of accelerations. If the driving force is constant, like gravity, the accelerations would be constant, or uniform. The graph of a = a(t) would be a horizontal line. Using integration, the graph of v = v(t) would be a sloping line. What would the graph of x = x(t) look like?

These are ideal situations. In reality, our apparatus has all sorts of random variations, frictions, and uncertainties. The raw graph of a = a(t) would not be a horizontal line, the graph of v = v(t) would not be a sloping line, etc. However, using statistical analysis, we can negate to some extent the vagaries of the equipment.

Task:

Equipment:

Procedure:

  1. Set up the apparatus as shown below:

Figure 1

  1. Drag the tape (by hand) through the apparatus for a measured amount of time.
    1. There is no switch, just plug it into the 120V outlet.
    2. Be sure to drag the tape at a constant speed.
    3. The number of dots and the pulling time is all you need from this action.
  2. Set up the apparatus as shown below:

Figure 2

  1. Using the 20g mass, allow it to drag the tape through the tape timer. Be sure there is enough tape to reach the floor.
  2. Lay the tape on a meterstick and record the POSITION of (not the distance between) each mark.
  3. Repeat 3-5 for the 100g mass.

Spreadsheet:

  1. Title your file and label columns with units!
  2. Make a single cell for time/dot from the first procedure (pulling the tape by hand), then refer to this cell when you need t.
  3. Make a column for the raw data, the position of each mark.

  4. Next to this column, calculate the difference between positions.
    1. e.g. B2:(A2-A1), B3:(A3-A2)
  5. Next to this column, calculate the quotient of X/t.
    1. X is the difference between positions.
    2. t is the time interval derived from the 2) procedure.
      1. Find t by dividing the "pulled" time duration by the number of dots during that duration.
      2. You should have this number calculated somewhere in your sheet.
    3. these are average or midmark velocities
  6. Next to this column, calculate the difference between average velocities (V).
  7. Next to this column, calculate V/t. These are your accelerations.
  8. Make a column for elapsed time.

    1. This could be made using multiples of the incremental time.
    2. These values will be used for graphing.
  9. Compute the average acceleration. This is your first value for acceleration.

Graphs for 20g Mass:

Be sure to title, label and grid your graphs!

  1. Acceleration vs time:
    1. Type of graph: X-Y
    2. X: elapsed time
    3. A (first series): computed accelerations
    4. Trendline: which (line, power, polynomial, exponential) will you choose? Show the equation and the intercept on your chart.
      1. For Series 1 select the series by right-clicking on a datum. Under Chart Tools / Layout choose the Error Bars pull down menu.Choose Error Bars with Standard Deviation.
      2. Right click on any x error bar to delete them all.
      3. Y error bars are a good way of showing data variation around a specified value.
    5. Your second value for acceleration lies here.
  2. Velocity vs time
    1. Type of graph: X-Y
    2. X: elapsed time
    3. A: computed velocities
    4. Trendline: which will you choose? Show the equation and intercept on your chart.
    5. Your third value for acceleration lies here.
  3. Displacement vs time
    1. Type of graph: X-Y
    2. X: elapsed time
    3. A: displacement (raw data)
    4. Trendline: which will you choose? Show the equation and the intercept on your chart.
    5. Your fourth value for acceleration lies here.
  4. Repeat for 100g mass. You will have 6 graphs in your report.

Questions:

  1. Full analysis on what affects this experiment: Why such great variations in results? How certain are you about your data? Your calculations? Use your understandings from the Uncertainty, Statistics and Measurement exercise.
  2. On the v(t) graph, what does the y intercept represent?
  3. What do the quadratic coefficients b and c represent?
  4. Make a table (below) and compare the four values for acceleration for each of the two masses. Which is the most trustworthy (and by this I DON"T mean closest to g)?
  20g 100g
average    
accel. graph    
vel. graph    
displace. graph