Ball and Ramp Projectile Behavior

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data program

Caution:  Throughout this experiment, avoid allowing the steel ball to fall on any valuable surface that might be dented or damaged.  This would include but is not limited to unprotected tile, hardwood and vinyl floor surfaces.  Use common sense and if you prefer to do this experiment on such surfaces, take appropriate measures to protect them.

If you set up a ramp near the edge of a table and allow a ball to roll down the ramp, off the edge and fall freely to the floor, you can, within the limits of experimental uncertainty, tell how fast the ball was going from the angle of the ramp, the distance the ball falls and the height of the ramp from the floor.

As long as the ball is traveling reasonably fast, and as long as the ramp isn't so steep that the ball begins slipping as it rolls, the experimental uncertainties in this experiment are low, so we can with a little care obtain very accurate results.

Using this setup we can test the hypothesis that acceleration is uniform on the ramp, we can measure the acceleration of gravity with good accuracy, and we can test the hypothesis that the horizontal and vertical motions of a projectile are independent.

Copy this document, from this point to the end, into a word processor or text editor. 

The basic activity here is to allow a ball to roll off a gently inclined ramp and fall to the floor, observing its horizontal range.. 

As the ramp gets steeper the landing position of the ball changes. 

You are going to see how far the ball travels in the horizontal direction, after leaving the ramp, for three different slopes. 

Then you'll use a complicated formula to figure out how fast the ball was traveling as it left each ramp, and you'll use these results to figure out the corresponding accelerations. 

The ultimate goal is to figure out how the acceleration of the ball changes with the slope of the ramp.

You will set up the longer of the two grooved ramps in your lab kit as shown in the picture below, on the edge of a table or counter between half a meter and a meter above a level floor. 

Two sheets of paper, as shown, will be placed on the floor (perhaps with the rectangular piece of plywood that came with your kit used as a backing to protect your floor) so that when the steel ball rolls down the incline and off the end, it will fall freely to the floor and strike the paper.  The small piece of carbon paper that came with your kit will be placed on top of the paper, giving you a clear mark on the paper when the ball strikes.  The carbon paper might be stapled inside the pack of copies of paper rulers; if it is not there it should be clearly visible when you unpack your lab kit.

The system should be within reach of your computer, so you can make some timing measurements.

 

 

Observe what happens when a ball is released from the top of the ramp:

Determine where a ball dropped from just below the edge of the ramp lands:

Determine where a ball allowed to roll down the ramp and to fall freely to the floor will land:

Your paper will look something like the one pictured below, with 5 marks clustered at the straight-drop point and the other at the projectile-landing point..

 

Measure the positions of the marks:

 

The measurements you report here must be in centimeters.  If your measurements need to be converted to standard cm, please do so.

In the space below, in comma-delimited format report in the first line the five positions resulting from straight drops. In the second line report the five landing positions of the ball after rolling down the ramp. In the third line report the vertical drop.  Again, be sure all measurements are in cm. 

Starting in the fourth line briefly explain how you obtained your measurements.

-------->>>>>>> straight drops, landing positions, vertical drop

Your answer (start in the next line):

 

 

#$&*

Find the mean and standard deviation of your five straight-drop positions and your five landing position:

Report the mean and standard deviation of the straight drop positions in the first line of the space below, and the mean and standard deviation of the 5 landing positions in the second line, using comma-delimited format in each line. 

Starting in the next line briefly indicate how you obtained your results.

-------->>>>>>>>>> mean & sdev straight drop, same for landing positions

Your answer (start in the next line):

 

 

#$&*

 

You will now place your taped-together sheets of paper back in the original position, add first one then two dominoes to the stack, and obtain something that looks very much the pictures shown below.  Specific instructions follow the pictures.

Add another domino to the stack (the stack will now contain 3 dominoes) and obtain 5 more marks for the new ramp slope.

Report the positions of the five new marks, in comma-delimited format on the first line below.  Starting in the second line give a brief explanation of the meaning of your data and how they were obtained.

---------->>>>>>>> 3 dominoes five new positions, explanation

Your answer (start in the next line):

 

 

#$&*

 

Repeat the procedure once more with a fourth domino added to the stack, obtaining 5 new marks.

Report the positions of the five new marks, in comma-delimited format on the first line below:

---------->>>>>>>> 4th domino 5 positions

Your answer (start in the next line):

 

 

#$&*

 

You will now determine for each setup the mean distance traveled by the ball after leaving the ramp:

The mean distance traveled by the ball for each setup will be the magnitude of the difference between the mean straight-drop position and the mean projectile-landing position.

Starting in the seventh line give a brief explanation of the meaning of your numbers and how they were obtained.

---------->>>>>>>> mean std dev landing positions 2 dom, same 3 dom, same 4 dom, mean, mean horiz dist 2 dom, same 3 dom, same 4 dom

Your answer (start in the next line):

 

 

#$&*

 

From the angle of the ramp, the vertical drop and the horizontal distance traveled by the ball after leaving the ramp, it is possible to use the equations of uniformly accelerated motion to determine the velocity of the ball as it left the ramp. 

First do an approximate calculation, based on the not-quite-accurate assumption that the ball is moving only in the horizontal direction as it leaves the ramp, with 0 velocity in the vertical direction:

---------->>>>>>>> time of fall, explanation

Your answer (start in the next line):

 

 

#$&*

 

---------->>>>>>>> horiz vel each setup, explanation incl sample calc

Your answer (start in the next line):

 

 

#$&*

 

The above analysis was not completely accurate, since the vertical velocity of the ball was not actually zero in any case.

Answer the above series of questions in the space below:

---------->>>>>>>> effect of nonzero vertical velocity

Your answer (start in the next line):

 

 

#$&*

 

The complete analysis of the motion of the ball follows. 

General College Physics and University Physics students will be expected to understand the analysis; Principles of Physics students should understand the general scheme and are encouraged, but not required to understand all the details of this analysis.

You can skim (skim, not skip) this analysis now and come back to it after completing the experiment.

The ball leaves the ramp with an unknown velocity, which we will call v, at an angle theta below horizontal.  We can easily determine theta from the slope of the ramp (tan(theta) = ramp slope). 

The ball therefore has an initial downward vertical velocity v0_y = v sin(theta) and initial horizontal velocity v0_x = v cos(theta).

Let clock time be t = 0 at the instant the ball leaves the ramp.  The horizontal velocity is constant so at a later clock time t the x position of the ball will be

Let the downward direction be positive.  Then the initial vertical velocity is in the positive direction, as is the acceleration of gravity, so the vertical position at clock time t will be

v0_x and v0_y can both be expressed in terms of the unknown initial velocity v and the known angle theta. 

At the instant of impact, x will be equal to the horizontal range of the projectile, and y will be equal to its distance of fall.  Using x_range and y_fall for these distances we have the two equations

Writing v cos(theta) and v sin(theta) for v0_x and v0_y the equations become

All quantities in these equations are known, except t and v.  So we have a system of two simultaneous equations which can be solved for v and t.  (note that though we haven't yet solved for theta, we could do so at any time, given the information we have for the slope)

The solution is fairly straightforward.  We solve the first equation for t, obtaining t = x_range / (v cos(theta) ).  Then we plug this into the second equation to obtain

which simplifies to

You should be able to write this equation in standard mathematical notation, and when working through this analysis for yourself you should do so.  For easy reference the equation looks like this:

It is straightforward to solve this equation for v.  We obtain

Since we are only interested in the speed of the ball, we use the positive solution.

In standard simplified form the positive solution would be represented as follows::

We still haven't figured out theta. 

  • sin(theta) = ramp slope / sqrt(1 + ramp slope^2) and
  • cos(theta) = 1 / (1 + ramp slope)^2.

Once we have this information we can plug g, x_range, y_fall and our theta-related information into the equation to get v.

This formula will probably fairly meaningless to most students at this point, though most should have a fair idea of how it was obtained.  The calculation has been built into the data analysis program.  Just click on the Experiment-Specific Calculations button and choose the Ball and Ramp Projectile Experiment. Enter the following upon request:

Your information and the velocity of the ball, in cm/sec, will appear in the data window.

&#&# The program works, if you enter the data as indicated.   If you get an error message, then give it another try, being careful to enter the data using the required syntax.  If it doesn't work, copy the contents of each text box used at each step, as it appears just before you click the button, and also specify which button you clicked.   Use one line for each textbox entry.  

If it worked, you can leave the space below blank.&#&#

-------->>>>>>>> details, in case the program doesn't appear to have worked

Your answer (start in the next line):

 

 

#$&*

 

Use this feature of the program to determine ball velocity information for the first ramp, with two dominoes in the stack. 

Report these three velocities in the space below, in comma-delimited format on the first line.  Starting in the second line explain how your velocities were obtained, including a reasonably detailed sample calculation for at least one trial.

-------->>>>>>>> three velocities 2 dom (mean - std, mean, mean + sdev), explanation & sample calc

Your answer (start in the next line):

 

 

#$&*

 

Now repeat this series of three calculations for the 3-domino trials, and report your three velocities in the space below, in comma-delimited format on the first line.  You need not include the detailed calculation, provided it was done in the same manner as the one you reported in the preceding space .

-------->>>>>>>> same 3-dom

Your answer (start in the next line):

 

 

#$&*

 

Repeat once more for the 4-domino stack, and report your three velocities in the space below, in comma-delimited format on the first line.  (Again you need not include the detailed calculation, provided it was done in the same manner as the one you reported previously).

-------->>>>>>>> same, 4-dom

Your answer (start in the next line):

 

 

#$&*

 

Based on the velocity results for the 2-domino stack, if we assume that the acceleration of the ball was uniform, what are the predicted acceleration and the range of accelerations on the 2-domino ramp? Report in comma-delimited format on the first line, followed by an explanation starting on the second of how you obtained your results, including one sample calculation.

-------->>>>>>>> accel, range of accel, 2-dom, expl with sample calc

Your answer (start in the next line):

 

 

#$&*

 

Give the same results for the 3-domino stack, in the same format. No explanation is necessary this time.

-------->>>>>>>> accel, range of accel, 3-dom

Your answer (start in the next line):

 

 

#$&*

 

Give the same results for the 4-domino stack, in the same format. No explanation is necessary this time.

-------->>>>>>>> accel, range of accel, 4-dom

Your answer (start in the next line):

 

 

#$&*

 

Using your results, plot a graph of predicted acceleration vs. number of dominoes.  Then sketch and estimate the slope of your best-fit line.

Starting in the third line give a brief explanation of the meaning of your numbers and how they were obtained.

------>>>>>>>>  accel vs. # dom:  slope and y int best fit, coord of 1st point, coord of 2d point used

Your answer (start in the next line):

 

 

#$&*

 

Now extend the points of your graph to form error bars, using for each number of dominoes the maximum and minimum values you obtained for accelerations on those ramps.

------>>>>>>>>  steepest slope, least steep with error bars, meaning and how obtained

Your answer (start in the next line):

 

 

#$&*

 

NOTE that the next experiment continues using this setup.  In that experiment you will simply roll the ball from rest for distances of 10 cm, then 20 cm down each incline, with 5 rolls at each distance for a total of 10 rolls; this will be done for a series of 4 ramp setups.  You will also time 10 rolls for each of three different setups.  It should take you no more than 20 minutes to obtain the necessary data.  You might wish to go ahead and at least get that data now while you have everything set up, using the form Uniformity_of_Acceleration_for_Ball_on_Ramp.  However if you don't have time right now, the setup is simple enough and it shouldn't take you long to set this up again.

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: 


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Revised: 11 Apr 2014 10:34:14 -0400