Uniformity of Acceleration for Ball on Ramp

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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.

Most students have reported reasonable completion times in the range of 1 to 2 hours, but a significant number have reported unreasonably long times to complete this experiment. Since the major goal of the experiment is to test the uniformity of acceleration on a ramp, it has been decided that a single ramp slope will suffice. To reduce the amount of data by half, the instructions for this experiment, which originally required measurements on using both 2- and 4-domino stacks, should be modified as follows:

If your access code ends with an even digit then follow the instructions only for the 2-domino setups. If your code ends with an odd number follow the instructions for a 4-domino setup. Respond only to the questions that correspond to your selected setups; boxes requesting responses for the other setup may be left blank.

Set up the system as before, using the 2-domino stack or the 4-domino stack, according instructions in preceding labs.
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 3 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.

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

Also conduct a total of 10 timings:

Conduct 5 timings for your setup 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 (i.e., move the dominoes to the other end, which will reverse the downward direction)..

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 3 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 below.

---------->>>>> 10 cm 2 dom

Your answer (start in the next line):

#$&*

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

---------->>>>> 20 cm 2 dom

Your answer (start in the next line):

#$&*

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

---------->>>>> 30 cm 2 dom

Your answer (start in the next line):

#$&*

Flip a coin. If it comes up 'heads' then do the next step. If it comes up 'tails' skip the next step.

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

---------->>>>> 10 cm 2 dom rev

Your answer (start in the next line):

#$&*

If you got 'heads' when you flipped the coin, do the next step. If it was 'tails' then skip.

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

---------->>>>> 20 cm 2 dom rev

Your answer (start in the next line):

#$&*

If you got 'heads' when you flipped the coin, do the next step. If it was 'tails' then skip.

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

---------->>>>> 30 cm 2 dom rev

Your answer (start in the next line):

#$&*

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

---------->>>>> 10 cm 4 dom

Your answer (start in the next line):

#$&*

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

---------->>>>> 20 cm 4 dom

Your answer (start in the next line):

#$&*

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

---------->>>>> 30 cm 4 dom

Your answer (start in the next line):

#$&*

If you got 'heads' when you flipped the coin earlier, then you can skip. If it was 'tails' then do the next step..

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

---------->>>>> 10 cm 4 dom rev

Your answer (start in the next line):

#$&*

If you got 'heads' when you flipped the coin earlier, then you can skip. If it was 'tails' then do the next step..

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

---------->>>>> 20 cm 4 dom rev

Your answer (start in the next line):

#$&*

If you got 'heads' when you flipped the coin earlier, then you can skip. If it was 'tails' then do the next step..

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

---------->>>>> 30 cm 4 dom rev

Your answer (start in the next line):

#$&*

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.

---------->>>>>

Your answer (start in the next line):

#$&*

For each set of 5 trials you will now need to calculate the velocity of the ball as it comes off the ramp. There are various ways to calculate the velocity and its uncertainty. Here the velocity calculation for each setup will be based on the mean horizontal distances.

First calculate the mean horizontal distance traveled by the ball after leaving the end of the ramp, for each starting position on each setup (you had 3 starting positions for each setup).

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.

You will have skipped either the second or fourth setups. Just type 'skipped' for that line.

---------->>>>> mean horiz dist for each of 4 steps

Your answer (start in the next line):

#$&*

Using the 'Experiment-Specific Calculations' button of the data program, 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.

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:

For the setup you skipped, just enter 'skipped'.

---------->>>>> ball speeds based on mean distances

Your answer (start in the next line):

#$&*

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 the 2-domino 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 and also explain what units resulted from your calculation:

---------->>>>> accel 10 cm 2 dom setup, explain

Your answer (start in the next line):

#$&*

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 distance the ball moved down 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 below, three accelerations for each setup, in the same order and the same format used in the preceding box. Report three accelerations per line, separated by commas. You will report 4 lines, including one you can report as 'skipped'

---------->>>>> report accelerations

Your answer (start in the next line):

#$&*

You can copy the contents of the previous 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 (for example, you have three accelerations on the first ramp; you will find the mean and standard deviation of these three quantities).

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. Then starting in a new line briefly explain what your results mean and how they were obtained.

---------->>>>>

Your answer (start in the next line):

#$&*

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.

---------->>>>> uncertainties in accelerations, how estimated

Your answer (start in the next line):

#$&*

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?

---------->>>>> support or reject hypotheses

Your answer (start in the next line):

#$&*

Report the 5 time intervals you obtained for the 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 system with the ramp reversed.

---------->>>>> 5 time intervals original orientation, *&$*&$

Your answer (start in the next line):

#$&*

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

Repeat for the setup with the ramp reversed.

Report your results below, each line consisting of the lower and upper limit for one setup, in comma-delimited format. You will have two such lines, which you should report in the order requested above.

---------->>>>>

Your answer (start in the next line):

#$&*

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?

---------->>>>>

Your answer (start in the next line):

#$&*

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 .5 seconds, then its horizontal velocity, which we assume to be uniform, is 20 cm / (.5 sec) = 40 cm/s.

Now answer the following:

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 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.

Your answer (start in the next line):

#$&*

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.

Do so for the present example, and report below as follows:

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.

---------->>>>>

Your answer (start in the next line):

#$&*

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?

---------->>>>>

Your answer (start in the next line):

#$&*

Now modify your answers to acknowledge the uncertainty in velocity. Instead of using vf for average velocity, use the 'mean +- uncertainty' expression vf +- h, where vf represents the mean and h the uncertainty in the final velocity. Continue to use `ds for the distance down the ramp.

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

---------->>>>>

Your answer (start in the next line):

#$&*

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 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 dv, 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.

Your response (start in the next line):

#$&*

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:

---------->>>>>

Your answer (start in the next line):

#$&*

Your instructor is trying to gauge the typical time spent by students on these experiments. Please answer the following question as accurately as you can, understanding that your answer will be used only for the stated purpose and has no bearing on your grades:

Approximately how long did it take you to complete this experiment?

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