Falling Bodies

(last edited December 29, 2011)

Dr. Larry Bortner

 Learning Objectives

Explain the parabolic time dependence of the distance a mass moves under constant acceleration, as well as the linear time dependence of its velocity.

Create an analysis spreadsheet from a blank Excel sheet.

 Purpose

To investigate the motion of a body under constant acceleration, specifically the motion of a mass falling freely to Earth.  To verify the parabolic time dependence of the distance fallen and the linear time dependence of the velocity.

 Background

We use a digital free fall apparatus to measure times that a mass falls a given distance. This apparatus consists of a launcher with a screw release, a large ball bearing, a touch sensitive landing pad, and a millisecond-resolution digital timer operated through the Science Workshop interface (Fig. 1).

 

Description: Figure 1a

Figure 1 Digital free fall apparatus.

 

The experiment consists of measuring the distance from the ball in the launcher to the landing pad, then timing the ball dropping this distance. As noted in the Measurement and Uncertainty experiment, there will be some randomness associated with measurement of the time. The experimental fall time varies because of non-uniformity of both the release method and the initial ball position. Thus we need to determine u{t} statistically.

 

This data gives you the distance x vs. time t curve. For a constant acceleration a, zero initial position and zero initial velocity, one of the equations of motion is

 

(1)                     

 

From basic algebra (Oh, no!) we know that this is a quadratic equation and that when we plot it, the resulting curve is a parabola.

 

There are two different ways to express the average velocity of the ball after it falls a certain distance from rest:


         One way is to take the average of the initial velocity v0 and the final velocity v. The ball isn’t moving when we first release it, so v0 is zero. Without friction, v is also the instantaneous velocity of any object that has fallen the same distance starting at a zero initial velocity. Then vav = (0+v)/2 = v/2 .


 

         The other way is to divide the distance traveled by the time it took: vav = x/t.


 

 

Equating the two, we get that

 

(2)                     

 

A second equation of motion of an object under a constant acceleration a is

 

(3)                     

 

So if we calculate the velocity from Eq. 2 and plot it against the corresponding time, we should get a straight line with a slope equal to the acceleration of gravity g, and a zero y-intercept (the ball is falling from rest).

 A note to clarify matters: 

We seek to experimentally verify the equations of motion under constant acceleration, specifically Eqs. 1 and 3. Note that there are two different expressions for the velocity of the falling ball. One of the expressions, Eq. 2, is derived independently of the equations of motion under a constant acceleration. We use Eq. 2 to calculate the experimental value of v. Eq. 3 and Eq. 1 come from the theoretical mathematical model that we are trying to prove.

 

How do you calculate the velocity in the Analysis? You have to use Eq. 2. If you use Eq. 3, you are using the equation to prove the equation; that is, you get perfect results but you don't prove anything. This is something you have to watch out for in any experimental endeavor. Be aware of the context of any equation given in the Background.

 Procedure

You need the following items:

*        meter stick and 2-meter stick

*        launcher with set screw release

*        ball bearing

*        landing pad

*        Science Workshop interface

*        computer

 

1.     Set up the following two tables on your data sheet:

 

for x=

 

u{x} (cm)=

 

Trial #

t (s)

 

and

 

xsuggested(cm)

xactual (cm)

t (s)

 


2.     Click on Data Studio Experiments> First Quarter> Falling Bodies to start the DataStudio program.

3.     Set the ball in the release mechanism.

a.     Push in the pin at the top so that the ball is nestled in the hole of the flexible metal strip, between the brass contact and the strip. (See Fig. 2.)

 

Description: ball in launcher

Figure 2 Putting the ball in the launcher.

 

b.     Lightly tighten the thumb screw to lock the ball in place. Make sure the landing pad is directly beneath the ball.

4.     Set the initial distance that the ball falls to between 5 and 10 cm, measuring from the landing pad to the bottom of the ball with a meter stick as in Fig. 3.

a.     Record the actual distance in cm.

b.     Record your estimate of the uncertainty of the measurement, u{x}. How dependent is your measurement on the viewing angle? When or where exactly does the ball bearing cause the circuit to be completed and the timing to stop?

 

Description: measuring ball drop distance

Figure 3 Measuring the ball drop distance.

 

5.     Click on the Start icon at the top of the DataStudio window.

6.     Untwist the thumb screw to release the ball.

7.     Record the Time of Fall.

a.     If the ball did not land on the pad, disregard this time (a single line through the number) and retake the data.

b.     Do not click on the STOP icon.

c.     Reseat the ball.

8.     Repeat Steps 6 and 7 until you have ten valid times.


 

       The timer automatically resets when you untwist the set screw after locking the ball.


 

       There should not be a spread of more than 0.010 s.


 

 

One of the many things to learn from the Measurement and Uncertainty experiment is that…

The standard deviation of any ten measurements of a single quantity measured in the same way is the uncertainty in any single measurement.

 

We assume that this error in the time measurement is independent of the time interval, meaning that it will be the uncertainty in the times measured at all distances. So your time uncertainty u{t} for this experiment for each timing is the standard deviation of these ten times. You still estimate an uncertainty of the time to record on your data sheet, but do not put any calculations on the data sheet. In the Error Analysis of your report, you state the calculated standard deviation as the actual time uncertainty.

 

9.     Make single time measurements for each of the suggested distances in the following table. They do not have to be exact. As long as you're within a few millimeters, you're OK.

a.     For the last distance, make it as large as you can put the landing pad on the floor and position the launcher as high as it will go and so that the ball will drop over the edge of the lab bench. Use the two-meter stick to measure the distance.

b.     After the measurements you should have at least 30 times; 10 times for the first distance and single times for the 20 subsequent distances.

           

x (cm)

x (cm)

x (cm)

x (cm)

21.6

33.1

47.1

63.5

23.7

35.7

50.2

67.1

25.9

38.4

53.4

70.8

28.2

41.2

56.6

74.5

30.6

44.1

60

(to the floor)

                                                                                   

10.  Click on the timer Stop button and exit DataStudio. Do not save the activity.

 Analysis

1.     There is no template for this experiment. Starting from the Excel Template> generic lab spreadsheet, copy in your first distance, the uncertainty u{x}, and the ten times in appropriately labeled cells. Remember to label the cells appropriately and to shade all data cells.


2.     Use the AVERAGE function of Excel to calculate tav for this distance.

3.     Use the STDEV function to calculate u{t}.

4.     Type these headings into your spreadsheet:

 

t (s)

x (cm)

v (m/s)

u{v} (m/s)

 

5.     Fill in the first two columns with your distance and time data. In the first row, the first time should be tav and the first distance the initial distance where you took the ten times.

6.     In the third column calculate the instantaneous velocity v in meters per second from Eq. 2. Note that this calculation includes a units conversion.

7.     In the fourth column, calculate the propagated error u{v} in the velocity from the following expression:

 

(4)                     

 

8.     Plot x vs. t. The distance x is along the y-axis and t is along the x-axis.

a.     Highlight the two columns of data values for t and x.

b.     From the Insert tab in the Charts group choose Scatter> Scatter with only Markers.

c.     From the Design tab of the Chart Tools in the Chart Layouts group, choose Layout 9. This layout automatically inserts text boxes for the chart and axes titles, as well as a linear trendline.

d.     Move the graph to the top of the spreadsheet and resize it, making it bigger and squarer.

e.     Remove the legend and give appropriate titles to the chart and axes.

f.      The line seems to fit the middle points OK, the endpoints not so much. Recall that our theoretical relationship between x and t (Eq. 1) tells us that these points should fall on a parabola. Let’s see if this is the case. Right click on the trendline and choose Format Trendline… from the popup menu.

g.     In the Trendline Options choose a Polynomial of Order 2 for the Trend/Regression Type, Forecast Backward 0.2 periods, and Display Equation on chart. (Display R-squared value on chart should be unchecked.)

h.     The x2 coefficient in the equation should be close to g, 490 cm/s2. How well do the points fit the parabola? Does the bottom of the parabola cup coincide with the origin?

i.      Include x- and y- error bars.

9.     Plot v vs. t with a linear trendline.

a.     Highlight the t-values. While holding down the Ctrl key, highlight the corresponding v-values.

b.     Click on Insert> Charts> Scatter> Scatter with only Markers.

c.     Click on Chart Tools> Design> Chart Layouts> Layout 9.

d.     Move the graph below the first and resize it so that they are about the same size.

e.     Right click on the trendline and Forecast Backward 0.2 periods. (Display R-squared value on chart should be unchecked.)

f.      Include x- and y-error bars. Chart Tools> Layout> Analysis> Error Bars> More Error Bars Options…

              i.    For the Vertical Error Bars, you have calculated a different u{v} for each v, so you have to enter all of these as Custom> Specify Value, for both the Positive Error Value and the Negative Error Value.

             ii.    From Chart Tools> Layout> Current Selection, change Series 1 Y Error Bars to Series 1 X Error Bars. u{t} is the same for all values of t, so this is a Fixed value.

10.  Do a least squares fit with errors of v vs. t (i.e., use LINEST to get the slope, y-intercept and errors in the slope and the y-intercept).

g.     Compare the slope with the accepted value of 9.80 m/s2 and the y-intercept with the expected value.

 Questions

1.     In Step 4 of the Procedure, why don’t we measure the distance to the center of the steel ball instead of to the bottom?


2.     Is the diver below correct in his calculation? If he steps off that board, how fast in miles per hour would he be going when he hits the water? Show your work.

 

Description: diver2

Is this guy really a whiz?