The Acceleration due to Gravity via Free Fall
This Lab investigates how the gravitational interaction moves objects. (i.e. Is the
velocity constant? Does the object accelerate? Is the acceleration constant? How does the
acceleration change? etc.) Please be familiar with the equations of motion for constant
acceleration.
Before coming to the lab study the online demonstration at
http://mail.mcm.edu/~bykov.tikhon/demos/physlets/lab4.html
It represents a ball moving with constant unknown acceleration (Note that it is not
9.8 m s 2 in this particular animation, so you really have to find it, not just guess). If you
right-click on the location of the ball at any time moment, 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 major steps you would undertake to determine the
unknown acceleration of the ball.
Now 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 direction of velocity and acceleration as the ball moves up and
down. Does velocity have the same direction all the time? Does
acceleration have the same direction all the time?
As you can see either from your short experiment or from the online demonstration,
both balls are moving so fast that it is almost impossible to perform any measurements.
However, in the case of the simulation you could play it step by step to analyze details of
this 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. The free-fall electric spark apparatus.
2. The motion sensor with GLX data collecting interface.
3. Video capture of the free-falling object.
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Each of these apparatuses will allow an object to be dropped while recording its position
at regular time intervals. Each group will be working with only one of these apparatuses.
After making several measurements with a real experimental apparatus you can
use MS-Excel to follow through the calculation and to plot both the measured values and
the 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 the free fall electric spark apparatus:
It consists of a timing mechanism which provides an electric spark at accurately timed
intervals of 1/60 of a second. This spark perforates the waxed paper, thereby marking the
position of the falling object at the instant of the spark. Make sure that the weight is not
swinging when you drop it. It is very important that the apparatus be level. Start the
timer and drop the weight.
• If you are using the motion senor with GLX interface
Set it up so you can collect position vs. time data. Adjust the sensor’s data collecting rate
to a high enough value, so you can collect a sufficient amount of data points before the
falling object hits the floor. Make sure that sensor is not “seeing” anything else besides
the falling object and the floor. Experiment with different falling objects and choose the
one that provides the best data set. Save your data and transfer the data file from the GLX
to the computer for further analysis with Excel.
• If you are using video camera:
Make sure that the timer is displayed in the window, so you know how much time passes
between the frames. You will be dropping a ball in front of a painted board and
measuring the distance it travels during each time interval as measured by the frame
advance of the camera.
We define S1 as the distance traveled during one interval of time, S 2 as the distance
traveled during two intervals, etc. (After n intervals, it has gone a distance of S n .) 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 S n and S n −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 d n +1 ∆t .
4. What should you graph if you want to show how ``the velocity after a given time"
changes as time goes on?
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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 = [(dn+1/∆t) – (dn/∆t)]
and the acceleration is a = ∆v ∆t .
From your experimental data determine the distances S1 , S2 , etc.
6. Compare techniques for measuring d n versus measuring S n .
7. How does the uncertainty accumulate if you measure d n compared to if you
measure S n ?
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. You will
be using these data in a later lab, so be clear and explicit about what the columns are.
Tabulate the data with the time in the first column and displacement ( S n ) in the second
column. Include the uncertainties. If you measured d n instead of S n , then enter d n and
in the next column calculate S n from d n .
9. In order to calculate an average value of your repeated displacement
measurements, you must type the values into your calculator. Computers are
designed to perform calculations repetitiously (i.e., to do the tedium for you).
Can Excel ease the tedium of averaging each set of numbers? How?
Plot a curve of the values of S vs. time. Draw the best curve possible through the data
points.
10. What shape does the curve have? Flat? Linear? Quadratic? Cubic?
Exponential?
11. If linear, what values do the slope and intercept have? What variables would the
slope and intercept represent?
12. 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.
13. What shape does the curve have? Flat? Linear? Quadratic? Cubic?
Exponential?
14. If linear, what values do the slope and intercept have? What variables would the
slope and intercept represent?
15. If quadratic, what values do the coefficients have? What variables would the
coefficients represent?
If the acceleration (on a v vs. t graph, ∆v ∆t is the “rise over run”) is constant, the data
points will lie along a straight line. The acceleration may be determined from the slope
of this best-fit line. Compute the value of the acceleration of gravity from your graph and
compare with other groups to determine the known value for Abilene.
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A second estimate of g 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. Carry out this
calculation and compare the result with the result from above and with the known value.
16. Are your individual accelerations clustered around a reasonable value? Should
they be?
17. Are any values surprisingly different? If so, can you explain why?
18. How large are the uncertainties on the individual accelerations?
19. Are your individual accelerations consistent? Should they be?
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.”
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