As on all forms, be sure you have your data backed up in another document, and in your lab notebook.

Submitting Assignment: Uniformity of Acceleration for Ball on Ramp

Your course (e.g., Mth 151, Mth 173, Phy 121, Phy 232, etc. ):

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http://vhcc2.vhcc.edu/dsmith/genInfo/labrynth_created_fall_05/levl1_15/levl2_51/dataProgram.exe

This experiment tests the hypothesis that the acceleration of a ball on a ramp does not depend on where the ball is on the ramp or how fast it is traveling.  Due to secondary effects that occur with this system, this hypothesis may or may not be supported by the data.

If you are continuing the setup from the preceding experiment, here are the brief instructions for collecting data.  You may simply collect the marks and a few timing results, then complete the analysis and submit the form later.

Note:  If you have only the 15-cm ramp, then use a single domino rather than two dominoes, and use distanced of 5, 10 and 15 cm rather than the 10, 20, 30 cm distances indicated in the instructions.  If you don't have the carbon paper, use the best method available to you to determine the horizontal ranges of the falling balls.

• Set up the system as before, using the 2-domino stack.
• Position the paper as before.  You can use the same paper and the same positioning method you used in the preceding experiment, so that the straight-drop locations remain valid.
• Roll the ball from a position 10 cm from the end of the ramp, so that it lands and makes a mark as in the preceding experiment.  Repeat this 5 times.
• Do the same for a 20-cm roll.
• Do the same for a 30-cm roll.
• Mark the 'top' end of your ramp so you know which end is which.
• Place a second sheet of paper on top of the first, aligned perfectly with its edges.  Reverse the ramp and repeat the above.
• Now repeat the entire process using the 4-domino stack, with 5 rolls at each of the 10-, 20- and 30-cm distances, with the ramp in its original position then with the ramp reversed.   The instructions originally asked you to observe the 4-domino system with the ramp reversed, but you may omit this part and ignore any questions related to the reversed 4-domino ramp.  If using the 15-domino ramp, you will use only 2 dominoes rather than the 4 specified here.

You can analyze this data later.  The analysis will be very similar to what you have already done.

Also conduct a total of 20 timings of the ball rolling down the entire ramp:

• Conduct 5 timings for the setup using the 4-domino stack (if you are using the 15 cm ramp, use only 2 dominoes in the stack), with the ramp in its original direction.  As accurately as possible, use the TIMER program to determine the time required for the ball to travel the length of the ramp.
• Then conduct 5 more timings with the ramp reversed.
• Follow the same instructions for the 2-domino stack (1-domino stack if using the 15 cm ramp).
• Save your results for use when you analyze these data.

The remainder of this form, which will include instructions for analysis, will be posted shortly.

To analyze your data, you may again find the data analysis program to be the quickest way to get results. However note that if you have sufficient skills with spreadsheets and/or computer algebra programs, you may at some point find that you might prefer to use them to perform some of the repetitive calculations.

First, find the positions of each of the marks newly made on your papers. Each position will be measured along a single axis, and will be relative to the origin point you have selected.

Report in the first line, in comma-delimited format, the first 5 positions, relative to the origin, obtained from the 10-cm trials for the 2-domino ramp.

If you place a copy of this line into the data window of the data analysis program and click the 'Change Rows To Columns' button the rows will change to columns and you can then click the Mean and Standard Deviation button to obtain the mean and standard deviation of these positions.

Do so and report the mean and standard deviation, in comma-delimited format, in the second line of the box.

Using the same format report the same information for the first 20-cm trials on this ramp:

Using the same format report the same information for the first 30-cm trials on this ramp:

Using the same format report the same information for the 10-cm trials on the reversed ramp, still using the 2-domino stack:

Using the same format report the same information for the 20-cm trials on the reversed ramp:

Using the same format report the same information for the 10-cm trials on the reversed ramp:

Using the same format report the same information for the first 10-cm trials on the 4-domino setup, with the ramp in its original orientation:

Using the same format report the same information for the first 20-cm trials on this ramp:

Using the same format report the same information for the first 30-cm trials on this ramp:

Using the same format report the same information for the 10-cm trials on the reversed ramp, still using the 4-domino stack:

Using the same format report the same information for the 20-cm trials on the reversed ramp:

Using the same format report the same information for the 10-cm trials on the reversed ramp:

Report here the mean and standard deviation of the straight-drop positions, which will be used with the mean and standard deviations of the landing positions your reported above to find horizontal projectile distance information in each set of trials.

For each set of 5 trials you will now need to calculate the velocity of the ball as it comes off the ramp. The velocity is in fact a range of possible velocities, but the calculation you do here will be only the velocity based on the mean distance.

First calculate the mean horizontal distance associated with each of your 12 trials.

In the box below report in the first line, in comma-delimited format, the mean horizontal distances for the 10-, 20- and 30-cm trials for the 2-domino setup with the original ramp orientation.

In the second line use the same format to report the same information for the 2-domino setup with the reversed ramp orientation.

In the third line report for the 4-domino setup with the original ramp orientation.

In the fourth line report for the 4-domino setup with the reversed ramp orientation.

Using the 'Experiment-Specific Calculations' button, select 1, as you did in the preceding experiment, and respond with the information necessary to calculate the speed of the ball at the end of the ramp, based on the mean distance observed for your first set of 5 trials.  Note:  If you are using the 15-cm ramp, report double the number of dominoes you actually used.  (The program is set based on a 30-cm ramp and it uses the number of dominoes to determine the slope of the ramp).

Repeat this process for each of the remaining 11 trials.

Report the resulting speeds in the box below, three speeds for each setup and ramp orientation, in the same order and the same format used in the preceding box:

Each speed should match the final velocity of the ball on the ramp. In each trial the ball rolled a known distance from rest. Our hypothesis is that the acceleration was uniform and depended only on the slope of the ramp.

Assuming uniform acceleration from rest through a 10-cm distance, with final velocity equal to that you just reported for this setup, what would be the acceleration of the ball? Report that acceleration in the first line below, and starting in the second line explain how you obtained it:

You could easily and quickly repeat these calculations for the remaining 11 trials, and may do so if you wish. However, first click again on the 'Experiment-Specific Calculations' button, select 2, and enter when prompted the displacement of the ball on the ramp (10, 20 or 30 cm as required), the velocity reported previously for the trial, and the program will indicate the associated acceleration.

Using whatever means you believe will be most efficient, calculate the remaining accelerations.

Report the resulting accelerations in the box below, three accelerations for each setup and ramp orientation, in the same order and the same format used in the preceding box:

You can copy the contents of the box into the data analysis program, change rows to columns, and calculate the corresponding mean and standard deviation of the three accelerations reported for each of the four setups.

Using the program or another means of your choosing, calculate and report these results in four lines, in each line giving the mean and standard deviation of the acceleration.

What are the uncertainties in the information used to calculate distances, velocities and accelerations in this analysis?

How much uncertainty do you think there is in the acceleration results?

Specifically, what do you believe to be the percent uncertainty in the accelerations calculated here? That is, give the typical uncertainty as a percent of the typical acceleration. As best you can, explain how you can determine these uncertainties based on uncertainties in the original data.

Do your results appear to support the hypothesis that acceleration is independent of velocity or position on the 2-domino setups?

Do your results appear to support the hypothesis that acceleration is independent of velocity or position on the 4-domino setups?

Report the 5 time intervals you obtained for the 2-domino system in its original orientation, in comma-delimited format on the first line. Report the mean and standard deviation of those time intervals in comma-delimited format on the second line.

Then using similar format report on the third and fourth lines the same information for the 2-domino system with the ramp reversed.

Continue the pattern established above for the 4-domino system in its original, then its reversed orientation.

Based on `dt = mean - standard deviation then on `dt = mean + standard deviation, calculate the upper and lower limits on the acceleration in the 2-domino setup, with the ramp in its original position.

Repeat for the 2-domino setup with the ramp reversed.

Repeat for the 4-domino setup with the ramp in its original position.

Repeat for the 4-domino setup with the ramp reversed.

Report your results in the box below, each line consisting of the lower and upper limit for one situation, in comma-delimited format. You will have four such lines; report them in the order specified here.

Are these acceleration ranges consistent with the results previously obtained for the 10-, 20- and 30-cm trials on the corresponding ramps? Would you have more faith in the results of your timing or in the results obtained from projectile behavior?

Principles of Physics students may stop here.

General College Physics and University Physics students should continue:

Accelerations in this experiment were calculated based on observed distances, ramp angles and the acceleration of gravity.

Let us for the moment assume that ramp angle and acceleration of gravity contribute much less to the experimental error than uncertainties in the distances traveled by the projectiles.

Suppose that the horizontal distance traveled by a projectile is 20 cm with a 2% uncertainty. 2% of 20 cm is .4 cm, so the actual distance would be between 19.6 cm and 20.4 cm.

If the time of fall for this projectile is 4. seconds, then its horizontal velocity, which we assume to be uniform, is 20 cm / (.4 sec) = 50 cm/s.

What is the range of possible velocities, assuming the actual distance to be within 2% of the 20 cm mean? Report as two numbers in the first line of the box below, one number indicating the least possible and the other the greatest possible result.

In the second line report the same information for the velocities, but in the form mean + - uncertainty (e.g., if we were reporting the 20 cm distance with its 2% uncertainty the format would be 20 +- .4 cm).

In the third line report the uncertainty in velocity as a percent of the velocity.

Starting in the fourth line explain how you obtained your results.

Now suppose you are using this velocity to calculate the acceleration of a ball which rolled 30 cm from rest, ending up at this velocity.

The process could be carried out as followed:

• You might first take half of the final velocity as the average velocity, since the ball started from rest and accelerated in what we assume is a uniform manner.
• Then dividing this result into the displacement you would find the time required to roll down the incline.
• Dividing this into the final velocity, which since we start from rest is the change in velocity, we would obtain the acceleration.

• In the first line give the time interval obtained by using the lowest and the highest of the ball velocities reported previously.
• In the second line give the time range in the form mean +- uncertainty.
• In the third line report the percent uncertainty in the time interval.
• In the fourth line report the result of dividing the least of the velocities in your velocity interval by the greatest of the possible time intervals, and the result of dividing the greatest of the velocities in your velocity interval by the least of the possible time intervals.  These results will indicate the range of possible accelerations.
• In the fifth line report your acceleration results in mean +- uncertainty format.
• In the sixth line give the percent uncertainty in your acceleration.

Symbolic Analysis of Errors

You can use a formula to determine the uncertainty in acceleration based on the uncertainty in the final-velocity measurement:

If distance down the ramp is `ds and final velocity is vf, then assuming uniform acceleration from rest:

• What is the expression for the average velocity?
• What therefore is the expression for the time `dt required to roll down the ramp?
• What therefore is the expression for the acceleration?

If your final velocity is given symbolically in 'mean +- uncertainty' format as vf +- h, where vf might represent the mean and h the uncertainty in the final velocity, and the distance down the ramp is `ds, then

• What is the corresponding expression, in the form mean +- uncertainty, for the average velocity?
• What is the resulting expression for the time down the ramp?
• What is the corresponding expression for the rate of change of velocity with respect to clock time?
• Simplify this expression to show most clearly the relationship between the uncertainty in the final velocity and the resulting uncertainty in the acceleration.

The following is for all University Physics students.  Ambitious students in Principles of Physics or General College Physics who also have a background in calculus are also invited to participate.

• For simplicity of expression we will let final velocity be represented by v, displacement down the ramp by s and time down the ramp by t.  In the following we will obtain the differential expressions ds and dv, which should not be confused with the `ds and `dv used in this course within the equations of uniformly accelerated motion. 

• In the previous context `ds and `dv represent change in position and change in velocity over a specific finite time interval `dt.  That notation is used to continually remind beginning students that these quantities are always observed over specific intervals, which tends to avoid one type of common confusion among beginning students.  However it does tend to create confusion with differential notation, and you have to be careful to remember in which context you are working.

• Using the more economical notation we have a = 1/2 * v^2 / s.

What is da / dv?  i.e., calculate the derivative of the right-hand side of this equation with respect to v.

What is da / ds; i.e., calculate the derivative of the right-hand side of this equation with respect to s.

da / dv can be interpreted as the rate at which your value for the acceleration a changes with respect to the value of v on which it is based. 

Previously you calculated values of a which were based on your upper and lower estimates of v, and you obtained a corresponding range of possible values for a.

You should have obtained da/dv = v / s.  This can be rearranged (with apologies to the mathematicians and a note to be aware of the deeper mathematical meaning of the rearrangement) as

• da = v / s dv,

where dv is regarded as the uncertainty in velocity and da as the uncertainty in acceleration.

Previously you reported an acceleration result for a specific ramp in the format mean +- uncertainty, based on velocity results that were also in this form.

In the first line of the box below report your uncertainty in velocity and uncertainty you the obtained for acceleration.  Give two numbers in this order, in comma-delimited format.  These two numbers will have been reported in your previous responses.

Using the formula da = v / s ds, substitute dv for the uncertainty in velocity, v for the mean velocity and s for the displacement.  What do you get for da?  Report this in your second line.

Beginning in the third line discuss whether the results you reported here were consistent.

We could use the method of the preceding analysis for each setup, substituting the maximum and minimum velocities to find the minimum and maximum time intervals, then using those again with the velocities to obtain acceleration estimates, and finally inferring uncertainties in acceleration.

It's much easier to use the formula da = v / s dv.

Using the formula da = v / s dv, estimate the uncertainty in acceleration for each of the setups used in this experiment.

In comma-delimited form, report for each setup the number of dominoes, the mean velocity, the displacement down the ramp, the uncertainty in velocity, and the resulting uncertainty in the acceleration (based on the formula da = v / s dv).  Report one comma-delimited line, in the given order, for each setup:

You may add optional comments and/or questions in the box below.