2.2 The Acceleration due to Gravity via Free Fall
Pre-Lab activity
Please complete this activity before coming to the lab session and submit your results
through Moodle.
Consider the online simulation of a ball moving with constant unknown
acceleration (Note that this particular animation does not represent free fall, so you really
have to find the numeric value of unknown acceleration, not just guess). If you right-click
on the location of the ball at any time, its position will be displayed in the low left corner
of the window. Use these data as if you collected them in the real experiment and think
about the main steps you need to undertake to determine the unknown acceleration of the
ball.
To resolve this problem as well as to perform calculations for the real experiment,
which you will perform in the lab, you have to be familiar with the equations of motion
for constant acceleration.
Using MS-Excel may be handy to complete this task. Please recall what you have
learned about MS-Excel before attempting the data analysis.
After you have completed answering the online Pre-Lab question, examine the
motion of a real ball. Toss the ball a couple of times and observe its motion as it falls
down. Based on these observations predict the answers for the following questions:
1. Describe the motion of the ball in terms of its position, velocity and acceleration
as functions of time.
2. What can you say about velocity of the ball at the highest point of its trajectory?
3. What can you say about acceleration of the ball at the highest point of its
trajectory?
4. Think about directions of velocity and acceleration as the ball moves up and as it
moves down. Does velocity have the same direction all the time? Does acceleration
have the same direction all the time?
Lab Experiment
As you could see either from your short experiment or from the Pre-Lab online
simulation, in both cases the ball was moving so fast that it was almost impossible to
perform any measurements. However, in the case of the simulation you could play it step
by step to analyze details of motion. This step-by-step analysis helped you to obtain the
data for velocity and acceleration of the ball as functions of time and determine the
unknown acceleration of the ball.
A similar analysis can be performed experimentally using one of the following
methods and/or tools:
1. Free-fall electric spark apparatus.
2. Motion sensor with data acquisition interface.
3. Video capture of a free falling object.
Each of these apparatuses will allow an object to be dropped while recording its position
at regular time intervals.
Each student group will be working with only one of these apparatuses.
After making several measurements with real experimental apparatus you can use
MS-Excel to follow through calculation and to plot both measured values and calculated
values.
In this lab, the handout will be asking questions to guide your reasoning. Pay
attention to the types of questions asked. In future labs you will have to figure out the
questions as well as the answers. Please answer each question as you get to it before
reading further.
• If you are using free fall electric spark apparatus
• If you are using motion senor with computer interface
• If you are using video camera
The following ideas maybe useful as you work on the analysis of your data. Some of the
suggestions here may be useful for one or for another method of data collection.
We define x1 as the distance traveled during one interval of time, x2 as the distance
traveled during two intervals, etc. (After n intervals, it has gone a distance of xn .) We
define d1 as the distance traveled during the first interval, d 2 as the distance traveled
during the second interval, etc.
1. Write an equation for d n in terms of xn and xn 1 .
2. What should you graph if you want to show how “the displacement after a given
time” changes as time goes on?
3. What should you graph if you want to show how “the distance traveled per time
interval” changes as time goes on?
Therefore, d n t is the average velocity during the nth time interval. The average
velocity during the next time interval is dn1 t .
4. What should you graph if you want to show how ``the velocity after a given time"
changes as time goes on?
5. What should you graph if you want to show how ``the velocity change during a
time interval" changes as time goes on?
The
increase
in
the
velocity
during
a
time
interval
t
is
then
v = [(d n+1/t) - (d n /t)] and the acceleration is a  v t .
From your experimental data determine the distances x1 , x2 , etc.
6. Compare techniques for measuring d n versus measuring xn .
7. How does the uncertainty accumulate if you measure d n compared to if you
measure xn ?
8. Which can you measure more precisely? In the end, which is better?
We will now create an Excel worksheet which calculates and graphs these data. Be
clear and explicit about what the columns are. Tabulate the data with the time in the first
column and displacement ( xn ) in the second column. Include the uncertainties. If you
measured d n instead of xn , then enter d n and in the next column calculate xn from d n .
Plot a curve of the values of x vs. time. Draw the best fit curve possible through the
data points.
9. What shape does the curve have?
Flat?
Linear?
Quadratic?
Cubic?
Exponential?
10. If linear, what values do the slope and intercept have? What variables would the
slope and intercept represent?
11. If quadratic, what values do the coefficients have? What variables would the
coefficients represent?
If you haven't already, compute the values of d1 , d 2 , etc. and compute the values of
v1 , v2 , etc.
Plot a curve of the values of v vs. time. Draw the best fit (trendline) curve through
the data points.
12. What shape does the curve have?
Flat?
Linear?
Quadratic?
Cubic?
Exponential?
13. If linear, what values do the slope and intercept have? What variables would the
slope and intercept represent?
14. If quadratic, what values do the coefficients have? What variables would the
coefficients represent?
The acceleration can be determined from the one of the coefficients of the best-fit
line for the position vs. time graph. Compute the value of the acceleration due to gravity
from that graph.
The acceleration can be determined from the slope of this best-fit line of the
velocity vs. time graph. Compute the value of the acceleration due to gravity from your
graph.
Another estimate of acceleration can be made by calculating the average acceleration
between each pair of velocities (analogous to calculating the velocity from the
displacement) and then calculating an overall average of those individual accelerations or
by graphing that acceleration as function of time and fitting the trendline to that graph.
Carry out this calculation and compare the result with the result from above.
15. Are your individual accelerations clustered around a reasonable value? Should
they be?
16. Are any values surprisingly different? If so, can you explain why?
17. How large are the uncertainties on the individual accelerations?
18. Are your individual accelerations consistent? Should they be?
Compare your results with other groups to determine the known value for Abilene.
The phrase “compare these two measurements” means “calculate and discuss the %difference.” The phrase “compare your measurement with the known value” means
“calculate and discuss the %-error.”