Determining the Magnitude of Acceleration
First, find the instantaneous acceleration of the cart at a time during the constant
acceleration portion of the motion. Be sure you are on the “Acceleration” tab. Either use the
data in the table or let your cursor hover over a point on the acceleration graph and wait for
the corresponding ordered pair to pop up. Record the time and instantaneous acceleration
values below:
time = ________s instantaneous acceleration = _________m / s 2
Second, find the average acceleration of the cart. To the side of the data table, click on an
empty cell, it does not matter which one you choose. After clicking on an empty cell, type:
=average(
The equal sign lets Excel know that you want to make a calculation, and the function we are
choosing to use is the average function. Using both the acceleration data table and the graph
as a reference, determine a time frame during which the cart’s acceleration appears to be
most constant (include 5 to 10 data points).
With your cursor still blinking to the right of the opening parentheses, click on the
acceleration value that corresponds to the first value you wish you include in the average
calculation. Click and hold and drag down to the last acceleration value you wish to include.
Then type
)
to end the calculation. Hit enter. The average value should be displayed. Record the results
below.
For an example for how this looks on-screen, refer to Figure 3. This example is from an
entirely different set of data, but on the left you can see how the average function should be
entered and how the data should be selected, and on the right, how the result is displayed
once you hit enter.
Of course, you could always find the average by hand, but there is no time like the present to
learn a new skill, and having proficiency in Excel is a very helpful and transferable skill!
average acceleration: _________ m / s 2
range of times used in acceleration calculation: _________
Note that when acceleration is constant, the average and the instantaneous acceleration
should be the same!
Third, find the acceleration by calculating the slope of your velocity graph as described.
Click on the tab titled “Velocity”. Identify two points on the graph that lie in the time frame
during which the cart underwent constant acceleration. Move the cursor to the velocity graph
and hover over your first chosen point until the ordered pair pops up. Record the velocity and
corresponding time for that point under “Point 1” in Table 1. Use the same process for your
second chosen point and record the values under “Point 2”. For a more accurate answer, use
two points as far apart in time as possible but still during the time the cart was speeding up
with constant acceleration.
Use the points recorded in Table 1 to complete the calculations in Table 2.
The results from these three methods should agree within a few percent. If they do not, go
back and correct your errors.
Q15 Consider your results from the three different methods for calculating the cart’s
acceleration. How do you account for any slight numerical differences in your results?
Now we will explore one last method, employing the calculus you have been learning in
lecture. Click on the “Position for Fit” tab. Here we have trimmed the position data so that
the points where the cart was at rest are not included. This way we focus on the time during
which the cart was undergoing constant acceleration.
The graph of the trimmed position data has been created for you—now you will fit a
quadratic function to it (we choose quadratic based on the form of the kinematics equation
for position).
Click on the graph and a plus sign will appear at the top right of the graph. Click on the plus
sign and scroll down to “Trendline”, and then click on the dark triangle to the right of
“Trendline”. A menu appears—scroll down and click “More Options…”. For an example of
how all this should look on-screen, see Figure 4.
After clicking “More Options…”, a menu for formatting the trendline appears. Choose
“Polynomial”. The default is a polynomial of order 2; leave that since we want a quadratic.
Scroll down and click “Display Equation on chart”. The equation for the trendline now
appears on the graph.
Q16 Write down the equation for the position function. Keep in mind that Excel uses the
variables x and y by default for the input and output variables, respectively. Translate the
given equation such that t is the input variable and x is the output variable.
Q17 Determine the acceleration by taking two derivatives: the first derivative will give you
the velocity function, the second, acceleration. Enter them below. Does the acceleration
given by the second derivative of the trendline match the results from your previous
methods? It should! If it does not, go back and correct your error(s).
