Introductory Motion Experiments

Instructions: You are responsible for reading and following these instructions.
Experiment 2: Average velocity vs. slope for a ball on a single ramp; rate of change of velocity.

Experiment 3: The final velocity of a uniformly accelerating object released from rest is double its average velocity.
Experiment 4: Acceleration is constant for an object rolling down a uniform incline.
Experiment 5: Determination of the acceleration of gravity using acceleration vs. slope (ramp slope < .1).
Experiment 6: The horizontal range of a projectile with initial velocity in the horizontal direction is an accurate indicator of its initial velocity.

All experiments in this course are to be submitted electronically as Rich Text files, according to the following guidelines:

Except where inappropriate, all raw data are to be presented the form of clearly labeled tables, and are to be indicated as raw data. (Note: 'Raw data' refers to anything you observe directly, without doing any calculations of any kind).
All questions posed in the experiment are to be answered in self-documented form.
Procedures and analysis are to be described in self-documented narrative form.
When analysis involves repeated use of a calculation procedure, only a single example of that procedure is required. For subsequent uses of the procedure, it is sufficient to say something like 'using the same procedure we obtain the results inthe third column ... '
Wherever appropriate, significant intermediate calculations and results are to be presented in the form of clearly labeled tables.
Whenever appropriate, results are to be presented in the form of clearly labeled graphs.
Graphs are to be included in Microsoft Excel format, using a scatter plot (no other type of plot is appropriate to the graphing of most scientific information). If you with to use another format, specify that format and submit a sample for the approval of the instructor. Do not use bitmaps (e.g., graphs created using Paint or Paintbrush). You may if you wish scan neatly made handdrawn graphs into your document.
When you are asked to do something with a graph that is not easily accomplished with Excel or an approved alternative, describe in words what you did and what results were obtained.

Experiment 2: Measuring average velocity for a ball rolling from rest down a uniform track; Average velocity vs. slope; Inferring acceleration.

See also the video files related to experiment 3 for the idea of the setup.

We measure the time required for a ball to accelerate from rest down various uniform inclines with slopes between 0 and .1. For each incline we determine the average velocity of the ball. We then plot average velocity vs. slope and determine whether the graph appears to follow a straight line. Using the asserted propositions that the ball accelerates uniformly and that the final velocity of an object which accelerates uniformly from rest is double its average velocity, we determine for each incline the average rate at which the velocity of the ball changes. We then construct a graph of the average rate of velocity change vs. incline slope. We finally determine the slope of the straight line which best fit our data points.

Using the straight ramp (the back of the shelf standard balance) and the TIMER program (download from homepage; see videos), determine the time required for the ball to roll the length of the ramp, starting from rest, for several different ramp slopes:

Use about four different ramp slopes, with the four slopes more or less equally spaced and with the maximum slope about .1.

The slope of a ramp is its rise divided by its run.

Set the ramp on a level table (close to level is good enough for now) and measure the height of the top of the ramp at the elevated end, then measure the top of the ramp at the low end.

The rise is the difference in your measurements.

The run is the distance along the tabletop from where you made the first measurement to where you made the last.

Again, don't use a slope greater than .1; this means that you won't use a rise more that .1 times the length of the ramp.

Put your data into a table:

For each slope, write down the heights of the ramp at the two ends and the times required (you should time the ball at least three times for each slope).
The ramp length should also appear in your data report, as should anything else you measured.

Analyze the data:

Determine the average velocity for each slope, using the distance the ball traveled and the median time required.

Sketch a graph of average velocity vs. ramp slope.

Is the graph linear (that is, does it seem to stay close to some straight line)? If not describe how it varies from linear.

For a straight ramp, if the ball starts from rest (and not otherwise), we expect that the final velocity will be double the average velocity.

Using this fact, make a table of final velocity vs. ramp slope.

Using your table of final velocity vs. ramp slope, keeping in mind that since the ball starts from rest its initial velocity is zero, determine for each ramp the rate at which velocity changes.

Plot the rate of velocity change vs. ramp slope.

Fit the most appropriate straight line to your graph.
Determine the slope of this line.

Submit results, including

an organized data table
each of the tables you constructed
a detailed description of each graph and how well the straight line fits
complete and detailed, self-documented answers to all questions

Experiment 3: The final velocity of a uniformly accelerating object released from rest is double its average velocity.

By timing an object as it accelerates from rest down one uniform incline, then as it travels along a second slight incline elevated just enough to keep the object moving at a uniform velocity, we test the previously asserted hypothesis that when an object accelerates uniformly from rest, its final velocity will be double its average velocity. We do this by graphing the velocity on the second ramp vs. the velocity on the first. We interpret the slope of this graph to test our hypothesis.

Set up a two-ramp system as indicated on the video clip.

Be sure to use only the straight part of the curved-end ramp.

Carefully adjust the second (lower) ramp so that the steel ball will maintain a constant velocity, as indicated by the rhythm of your tapping as it crosses a series of equally spaced points (like the circular holes you will see in the ramp). If you tap the t key on the computer timer in this rhythm, the time intervals will tell you whether the velocity is pretty much constant.
Position the lower end of the first ramp a bit higher than the end of the second so that the ball rolls off of the first and onto the second ramp without losing significant velocity, and also without falling more than a millimeter or two.
Start with a slope such that the ball requires approximately 2 seconds to roll 20 cm down the first ramp to its end, starting from rest. Be sure to release the ball cleanly, without knocking it backward or pulling it forward.

Time the ball as it travels down the first ramp and an appropriate distance across the second, obtaining the time interval on each ramp. Use the computer timer program.

Time the ball from the instant of release to the instant it reaches the end of the first ramp, then for a measured distance across the second ramp.

The distance on the second ramp should be such that the ball requires approximately 3 seconds to travel the distance when the ball is rolled 20 cm down the first ramp.

This 2d-ramp distance should remain the same for all subsequent measurements.

It is suggested that you place an object for the car to bump into at the ending position on the second ramp; this will make your timing more accurate.

When timing, be conscious of your tendency to anticipate some events more than others (e.g., some people might anticipate the bump at the end more than the 'clank' from the first ramp to the second), and try to control the tendency.
Repeat your timing for the 20-cm distance two more times, so that you have 3 timings for this 20-cm distance.
Repeat for rolls of 10 cm, 15 cm, 25 cm, 30 cm, 35 cm, ..., 50 cm down the first ramp, until you reach the point where the ramp begins curving.
Present your data in a clearly labeled table.

Determine average velocities and plot a graph of your results:

For each distance, use the median (the 'middle') value of the time required to roll down the first ramp to determine the average velocity of the ball on that ramp.
For each distance, use the median value of the time required to roll along the timed portion of the second ramp to determine the average velocity of the ball on that ramp.
For each distance on the first ramp, plot the average velocity on the second ramp vs. the average velocity on the first.
Sketch a straight line which comes as close as possible, on the average, to your data points.

Don't worry about whether the straight line actually goes through any of the data points, worry about average closeness; a perfect graph of real data will rarely go through the precise position of of any data point.

Extend your straight line in both directions, slightly beyond the first and last data points.

Analyze your graph and answer questions:

By measuring the rise and the run of your straight line, determine its slope.
Does your straight line pass near the origin of your coordinate system? Why should it or should it not do so?
On which ramp does the ball have the greater average velocity? Why should this be so?
Which is the greatest and which is the least of the following: the average velocity of the ball on the first ramp, the initial velocity of the ball on the first ramp, or the final velocity of the ball on the first ramp?
Place in order, from the smallest to the largest, indicating any 'ties':

the average velocity of the ball on the

the average velocity of the ball on the second ramp,

the initial velocity of the ball on the first ramp,

the initial velocity of the ball on the second ramp,

the final velocity of the ball on the first ramp, and

the final velocity of the ball on the second ramp ( i.e., its velocity at the instant the timing ceased, not after you changed everything by stopping it).

Sketch a graph of the velocity of a ball vs. clock time from the instant it is released until the instant timing stops on the second ramp. Indicate the instant at which you think the ball reaches its average velocity on the first ramp.
Why might we suspect that the average velocity on the second ramp should be double that on the first, and how well does this experiment validate that hypothesis?

Analyze the potential effect of timing errors on your results for the 30-cm first-ramp distance (Principles of Physics students do only parts marked *)

* For the 30-cm distance, what do you think is the uncertainty in your measurement of the time interval on the first ramp (e.g., +-.01 sec, +-.05 sec, +-35 hours)? Why do you think that this is an appropriate value for the uncertainty?
* For your estimated uncertainty, we would expect that the actual time required was between the observed time minus the uncertainty and the observed time plus the uncertainty

For example, if the observed time is 2.3 seconds and the uncertainty is +-.02 sec, then we expect that the actual time lies between 2.3 - .02 sec and 2.3 + .02 sec, or between 2.1 sec and 2.5 sec.

We might say that the actual time `dt satisfies 2.1 < `dt < 2.5.\

In symbols, if the actual time is `dt, the observed time is `dt_obs and the uncertainty is t_unc, then we can say that `dt lies between the 'low estimate' `dt_obs - t_unc and `dt_obs + t_unc.

Written in the form of an inequality, we would say `dt_obs - t_unc < `dt < `dt_obs + t_unc.

* Based on your observed time and your expected uncertainty, what are the minimum and maximum values between which the actual time should lie?
* Had the actual time `dt been equal to the minimum expected value, then what would have been the average velocity on the first ramp?
* Had the actual time been equal to the maximum value, what would have been the average velocity on this ramp?
* What therefore do you expect is the minimum value that the actual average velocity might have had on this ramp? What is the maximum?
* On your last graph, near the point corresponding to the 30-cm distance, indicate this range of velocities as demonstrated on the video clip.

Repeat this process for the second ramp, and indicate the corresponding range of velocities as demonstrated on the video clip.
Block in the rectangle defined by your velocity ranges and see if your straight line actually passes through this rectangle.

Repeat this analysis for every data point on your graph (Principles of Physics students: repeat the analysis only for the 10-cm point and the 50-cm point).

For each data point, use the uncertainty in `dt that is appropriate to your means of timing for each point.
Does your straight line pass through all the resulting rectangles?
If not, is it possible to draw a straight line that passes through all the resulting rectangles?

We have analyzed the possible effect of timing errors.
We might also have analyzed the effect of errors in distance measurements.
Would you expect these errors to be more or less significant than timing errors, and why?

Submit results, including

an organized data table
a table of average velocity on the second ramp vs. average velocity on the first
a detailed description of your graph and how well the straight line fits
a report of the slope of your line and whether it seems, within experimental uncertainties, to pass through the origin.
complete and detailed, self-documented answers to all questions.

Experiment 4. Acceleration is constant for an object rolling down a uniform incline.

By timing an object as it accelerates from rest for various distances down an incline, we infer its average acceleration for various distances and average speeds. We then test to validate the hypothesis that the acceleration is in fact uniform lawn the incline.

You may use some of your data from the preceding experiment.

By releasing a ball to roll from rest through distances of 10, 15, 20, 25, 30, 35, 40, 45 and 50 cm down a uniform incline, you can determine whether the acceleration on the incline seems to be related to either the distance the ball rolls down the incline or to its average velocity on the incline.

Obtain (or use from your previous experiment, if you have sufficient confidence in your data) distance vs. time data for object rolling from rest down a straight incline similar to that used in the preceding experiment (unlike that experiment, you need not use the second ramp; if you use data from the first experiment, you will not need the data from the second ramp, and you will not report that data with this experiment).
Present your data in a clearly labeled and organized table.

Determine the average acceleration corresponding to each distance:

For each distance, use the median time to determine the average velocity of the ball for that distance.
Using the results of the preceding experiment (i.e., the relationship between average and final velocities on the first ramp), the average velocity and the fact that the ball started from rest, determine the final velocity of the ball for each distance.
Determine the change in velocity for each distance, then using the time required for each distance determine the average acceleration for each.

Graph acceleration vs. distance and draw conclusions:

Plot average acceleration vs. distance traveled on the first ramp and decide whether the acceleration changes with the distance in some systematic way, or whether the apparent variations in acceleration are simply the result of unavoidable experimental uncertainties.
How did you make use of the results of the preceding experiment to justify your calculation of the acceleration in this experiment?
Are the variations in acceleration observed in this experiment within the range of variations to be expected with the equipment you are using?
Do your results support the hypothesis that the acceleration of the ball rolling down a uniform incline is indeed independent of how fast the ball is rolling and of where the ball is on the ramp? If so, how do the results support hypothesis? If not, how do the results fail to support the hypothesis?

Analyze errors:

Using a procedure similar to that in the preceding experiment, determine the uncertainty in the acceleration for each point (Principles of Physics: analyze only the 10-, 30- and 50-cm points) and indicate these uncertainties on your graph as demonstrated on the video clip.
Is it possible to draw a straight horizontal line that passes between the error bars for every graph point?

Explain why, if it is so, then we have good support for the hypothesis that the acceleration on the ramp is constant.

Experiment 5. Determination of the acceleration of gravity using acceleration vs. slope (ramp slope < .1), and then by using time of fall as indicated by the pendulum timer.

For ramp slopes between 0 and .1, the acceleration of a cart down the slope is very nearly a linear function of ramp slope, with only a 1/2 % deviation from linearity within this range of slopes. In this experiment we determine the accelerations of the friction car corresponding to a selection of ramp slopes within this range, and plot acceleration vs. ramp slope. Since frictional forces and tabletop deviations from linearity will be very nearly the same for all trials, the slope of the graph gives us the acceleration of gravity.

Using the pendulum timer we can measure the time required for a dropped object to fall various distances from rest under the acceleration of gravity. We can calculate the acceleration for each distance and test hypothesis that the acceleration is the same for all heights.

Alternatively using the pendulum timer we can synchronize the dropping of a ball with the time required for the pendulum to complete a quarter-cycle for each of several drop distances. If the acceleration of the dropped object is uniform, then the time required for the object to drop should be proportional to the square root of the distance dropped. Since the period of the pendulum is proportional to the square root of its length, it follows that if gravity accelerates objects uniformly then the distance dropped should be proportional to the length of the synchronize to pendulum. From the associated proportionality constant, and from the proportionality constant for the pendulum period, we can determine from our data the acceleration of gravity.

Using the friction car on a series of small ramp slopes, you will first obtain data for the acceleration of the car vs. the ramp slope.

As shown on the video clip, ramp slope will be determined by measuring the length of the wooden ramp and its height above the supporting table at each of its ends.

For each ramp slope the acceleration will be determined by timing the car's motion over a fixed and measured distance from near one end of the ramp to the other.

Ramp slope will be changed by placing various objects under one end of the ramp.

Timing will be done using the pendulum apparatus, as indicated on the video clip.

The pendulum and the car will be released from rest simultaneously.

Ramp slope and/or pendulum length will be adjusted until the pendulum after an appropriate number of quarter-swings reaches the middle of its cycle at the same instant that the car collides with a barrier at the end of the ramp.

This means of timing eliminates some of the nearly unavoidable anticipation errors that tend to occur when an individual performs this experiment without assistance.

Using at least 5 more or less equally distributed ramp slopes between .04 and .1 (Principles of Physics students may use three such ramp slopes):

determine the length of the ramp and the heights of both of its ends, as measured from the table on which it rests
determine as accurately as possible the corresponding length and the number of quarter-swings of the pendulum that correspond to the time required to travel the measured distance
present your data in a clearly labeled table.

Determine the acceleration and ramp slope for each setup

For each ramp slope use your measurements of the heights of the ramp ends and ramp length to determine the slope of the ramp.
For each ramp slope use your T = A * L^p function from the pendulum experiment to determine the time interval corresponding to each quarter-cycle of the motion of the pendulum, and
use the number of quarter-cycles to determine the time required for the car to travel the measured distance.
Use the time required to accelerate down the ramp from rest, and the distance traveled during this time, to reason out the average velocity, final velocity and acceleration on each ramp slope.

Graph and interpret your results.

Construct an accurate graph of y = car acceleration vs. x = ramp slope.
Your graph points should lie near a straight line. Draw your best fit of the straight line and determine its x and y intercepts and its slope.
From the slope of the line, determine the change in acceleration that would correspond to a change of 1 in the slope.
For reasons you probably do not yet understand, if you have done the experiment correctly and accurately the slope of your graph should be equal to the acceleration that would be experienced by the car if it was simply dropped (or, equivalently, rolled down a vertical ramp).
This acceleration is called, for reasons that should be apparent, the acceleration of gravity.

Assess the quality of your results and analyze the potential effects of various potential errors on your determination of the acceleration of gravity. (Principle of Physics: * only)

* How accurately do you think you determined the time for each ramp slope? Describe how you made sure that your times were accurate and consistent.
* How close was your result to the accepted value of the acceleration of gravity, which is 980 cm/sec/sec?
* Exactly what is the significance of the point where y = car acceleration is equal to zero? What therefore is the significance of the x intercept of your graph?
What is the significance of the y intercept of your graph, which occurs where x = ramp slope is zero?
* Do the x and y intercepts of the graph have effect on your result for of the acceleration of gravity?
* Suppose that the table itself had a slope of .01, in the direction down the ramp.

Then every actual ramp slope on your table of acceleration vs. ramp slope should be .01 greater or less than the ramp slope you used on your table.

Which should it be, greater or less?

What effect would this have on your graph?

What effect would have on your result for of the acceleration of gravity?

Suppose that you had consistently misread the length of the pendulum, so that for every reading it was in fact 5 cm longer than you thought. How would this affect your graph, and what would be the effect on your result for of the acceleration of gravity?

Analysis of Errors (Principles students use only first, middle and last data points)

Using a procedure similar to that in the preceding experiments, determine the uncertainty in the acceleration for each of your final graph points and indicate these uncertainties on your graph in the usual manner.
Estimate the uncertainty in the measurements that you used to determine each ramp slope, and determine the range of possible ramp slopes for each data point. Indicate these uncertainties on your graph in the usual manner.
For each graph point sketch the rectangle defined by your minimum and maximum possible acceleration and your minimum and maximum possible ramp slope.
What is the slope of this steepest line that can pass through all these rectangles?
What is the slope of the least steep line that can pass through all these rectangles?
What therefore is the range of possible values of the gravitational acceleration, according to your experiment?

Experiment 6. The horizontal range of a projectile with initial velocity in the horizontal direction is an accurate indicator of its initial velocity.

In this experiment we measure the horizontal range of a ball projected horizontally from a ramp. We also measure the velocity of the ball at the end of the ramp by timing it as it rolls down the ramp from rest. Using the time required to fall to the floor we then determine the average horizontal velocity of the ball during its fall. We test the hypothesis that the horizontal velocity of the falling ball remains the same as at the instant the ball left the ramp.

When a ball accelerates down the curved-end ramp, with the curved end positioned horizontally at the edge of a table, the ball leaves the end of the ramp with a velocity which is in the horizontal direction, with no vertical component.

The time required for the ball to reach the floor is independent of its horizontal velocity, as we see from dropping a coin from the edge of a table with one hand as with the other hand we flick another coin off in the horizontal direction (the coins hit the floor simultaneously).

We can therefore use the distance of fall and the horizontal distance traveled to obtain the average horizontal velocity of the ball.

By intercepting the ball at various heights you will determine the horizontal range and therefore the horizontal velocity of the ball as it leaves the end of the ramp.

By timing the ball's motion on the ramp, you will determine its velocity at the end of the ramp.

First, using washers or coins as on the video clip convince yourself that the time required for an object to reach the floor when it initially travels in a purely horizontal direction is independent of its initial horizontal velocity.

Now set up the curved-end ramp.

Set up the ramp with the curved end right at the edge of a table. Press down on the curved end so that it is flat on the table, and support the other end to maintain this position.
Roll the ball down the ramp and see how, at the instant it leaves the end of the ramp, the ball is traveling in a horizontal direction.
Note also how the ball immediately starts falling, causing its path to curve downward.

Using books, chairs, buckets, and/or other common or uncommon household items, contrive to intercept the ball at various altitudes along its trajectory.

As demonstrated on the video clip, position a hardback book so that it intercepts the ball approximately 10 cm below the point where it leaves the ramp. The cover of the book should be horizontal.
Place a piece of notebook or typing paper on the book, and cover it with a carbonized sheet so that the ball will leave a mark on the paper when it strikes the book.
Measure the horizontal distance from the edge of the ramp to the edge of the paper. Note this distance on the paper, and note also the vertical distance from the bottom of the ball at the edge of the ramp to the book.
Then release the ball from the high end of the ramp and allow it to roll off the edge and freely fall until it strikes the book.
Reposition the paper to its original position and repeat twice more to obtain three horizontal distances.

Repeat for vertical distances of approximately 20 cm, 40 cm and 80 cm below the edge of the ramp.

Determine the time required to fall each distance, and use this time with the horizontal distance to find the average horizontal velocity for each vertical distance.

Using the distance fallen by the ball use the formula `ds = .5 g `dt² for a freely falling object falling from rest to determine the time required to fall each distance, using at `dt the median time observed for each distance.
Using the horizontal distance traveled by the ball during this time interval and the time required to fall, determine for each vertical distance the average horizontal velocity of the ball as it falls.

Plot y = average horizontal velocity vs. x = time of fall and use your plot to determine whether there is any experimentally significant dependence of average horizontal velocity on time or whether average horizontal velocity seems to be independent of time and therefore constant.

Do the differences in the horizontal velocities seem to be the result of random experimental uncertainties and errors in timing, or does there seemed to be a definite progression in the horizontal velocities as time increases?

By timing the ball for known distances on the straight part of the ramp, you can determine the velocity with which the ball leaves the ramp.

By determining the horizontal projectile distance (the distance traveled in the horizontal direction as the ball falls) corresponding to each distance on the ramp, you can use the time of fall to determine the average horizontal projectile velocity.

You can compare these results to test the hypothesis that average horizontal projectile velocity is equal to initial horizontal projectile velocity.

Time the ball from rest for distances of 20, 25, 30, 35, 40 and 45 cm along the straight part of the ramp, setting up a barrier at the approximate point where the ramp begins to curve.
Using these times and distances, determine the average velocity over each distance, and infer the final velocity for each distance.
For each trial, obtain data for the distance the ball falls and for its horizontal displacement after leaving the ramp.
For each trial, determine from `ds = 1/2 a `dt² the time `dt required to fall the observed vertical distance.
For each trial, use the time `dt obtained in the previous step and the observed horizontal displacement to determine the average horizontal velocity.

Graph y = final velocity on the ramp vs. x = average horizontal velocity after leaving the ramp. Assume that the ball gains very little velocity after reaching the curved end of the ramp.

Fit a straight line to your graph and determine the slope of this line.
If the horizontal velocity of the falling ball is constant, then why should we expect that the final velocity of the ball on the ramp should be equal to the average horizontal velocity of the falling ball?
If the final velocity of the ball on the ramp is equal to the average horizontal velocity of the falling ball, then what should be the slope of your graph?
If the horizontal velocity of the ball decreases as it falls, what should this do to the straightness of the graph and what effect should this have on the slope of the graph?
Answer the same questions if the horizontal velocity of the ball increases as it falls.
What conclusion do you draw from your graph regarding whether or not the horizontal velocity of the ball is constant?

Analysis of Errors

Estimate the uncertainties in your data, and indicate the results of these estimates on your graph in the usual manner.
Discuss the implications of these uncertainties.

Submit a report with all data and results in table form, with an explanation of each step of the analysis, and with all questions answered.