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Hi Expert,

My son is working on a roller coaster physics project. One of his experiments involved running a marble down a 30 foot length of PVC tubing with about a 1" diameter. In trial A, the top of the pipe (the starting point of the marble's journey) was located at a height of about 7 feet from the ground. The pipe had a straight slope to the ground. In the trial B, the pipe had the exact same starting point, but we introduced a curve into the track. Thus, the marble's initial descent down the track was steeper than in trial A. The steep drop was followed by a gentle uphill climb and a gentle final descent down the hill. The same exact length of pipe and the same marble was used in both trials.

As we understand it, a roller coaster's potential energy at the starting point of the first hill, the lift hill, is slowly lost over the course of the roller coaster's track through friction, wind, and braking. (In his experiment, we simulated a lift hill by simply starting the marble at the opening of the pipe that we held 80 inches off the ground).

Here is one explanation we read online:

"When a coaster is at the highest point of its track, it has high potential energy (energy of position). As the coaster accelerates down the hill, that potential energy changes into kinetic energy (energy of motion). Each time the coaster goes up another hill, the kinetic energy becomes potential energy again, and the cycle continues. Ideally, the total amount of energy would remain the same, but some is lost to friction between the wheels and the rails, wind drag along the train, and friction applied by the brakes. Because of this energy loss, each successive hill along a coaster track must be smaller than the previous hill in order for the train to continue along the course. " (http://www.essortment.com/roller-coaste ... 31074.html)

Our "coaster" did not have brakes or wind (the tube was enclosed), so only friction would affect the speed of the marble through the track, in theory.

We also ran another set of trials, C and D. In those trials, the pipe was kept straight (no curves / hills) but the slope of the track was changed. We had a low starting point (C, 30 inches off the ground) and a medium starting point (D, 55 inches).

For each trial, Matt ran the marble through the track about 10-12 times and recorded the speed through the track with a stop watch. As one might expect, the marble ran slowest through C and fastest through A. We think we understand why...what seems obvious is that the steeper slope makes you go faster because of the acceleration of gravity. My son says it's just like skiing, the steeper the run, the faster you go.

But here is what we can't quite figure out....why was trial B slower than A? The starting point height was the same. The marble was the same. The pipe used was the same. Wouldn't the friction then also be identical? The only difference was the curve in the pipe. Is there somehow less friction going uphill than downhill? In Trial B, in theory, the marble starts out faster since it has a steeper drop immediately. Then it slows as it goes up the hill and recovers speed as it goes down the hill to the end of the track. But adding it all up...why would the marble be slower in Trial B than A, since the friction is the same, the starting height the same and the marble and pipe the same.

What are we not understanding?

Thanks for your help!

Courtney and Matt

My son is working on a roller coaster physics project. One of his experiments involved running a marble down a 30 foot length of PVC tubing with about a 1" diameter. In trial A, the top of the pipe (the starting point of the marble's journey) was located at a height of about 7 feet from the ground. The pipe had a straight slope to the ground. In the trial B, the pipe had the exact same starting point, but we introduced a curve into the track. Thus, the marble's initial descent down the track was steeper than in trial A. The steep drop was followed by a gentle uphill climb and a gentle final descent down the hill. The same exact length of pipe and the same marble was used in both trials.

As we understand it, a roller coaster's potential energy at the starting point of the first hill, the lift hill, is slowly lost over the course of the roller coaster's track through friction, wind, and braking. (In his experiment, we simulated a lift hill by simply starting the marble at the opening of the pipe that we held 80 inches off the ground).

Here is one explanation we read online:

"When a coaster is at the highest point of its track, it has high potential energy (energy of position). As the coaster accelerates down the hill, that potential energy changes into kinetic energy (energy of motion). Each time the coaster goes up another hill, the kinetic energy becomes potential energy again, and the cycle continues. Ideally, the total amount of energy would remain the same, but some is lost to friction between the wheels and the rails, wind drag along the train, and friction applied by the brakes. Because of this energy loss, each successive hill along a coaster track must be smaller than the previous hill in order for the train to continue along the course. " (http://www.essortment.com/roller-coaste ... 31074.html)

Our "coaster" did not have brakes or wind (the tube was enclosed), so only friction would affect the speed of the marble through the track, in theory.

We also ran another set of trials, C and D. In those trials, the pipe was kept straight (no curves / hills) but the slope of the track was changed. We had a low starting point (C, 30 inches off the ground) and a medium starting point (D, 55 inches).

For each trial, Matt ran the marble through the track about 10-12 times and recorded the speed through the track with a stop watch. As one might expect, the marble ran slowest through C and fastest through A. We think we understand why...what seems obvious is that the steeper slope makes you go faster because of the acceleration of gravity. My son says it's just like skiing, the steeper the run, the faster you go.

But here is what we can't quite figure out....why was trial B slower than A? The starting point height was the same. The marble was the same. The pipe used was the same. Wouldn't the friction then also be identical? The only difference was the curve in the pipe. Is there somehow less friction going uphill than downhill? In Trial B, in theory, the marble starts out faster since it has a steeper drop immediately. Then it slows as it goes up the hill and recovers speed as it goes down the hill to the end of the track. But adding it all up...why would the marble be slower in Trial B than A, since the friction is the same, the starting height the same and the marble and pipe the same.

What are we not understanding?

Thanks for your help!

Courtney and Matt

- Courtney
**Posts:**42**Joined:**Thu Sep 18, 2003 5:36 pm

Well, you've stumped me, if that's any consolation. (And I used to teach mechanics at MIT long ago.) Your analysis is wrong, since it ignores the rotational kinetic energy of the rolling marble, but that still is not enough to explain your observations. For my curiosity, what were the actual times for case A and case B? I have one, rather strained hypothesis that might explain a small difference between A and B in the sense you observed: when the ball hits the top of the bump it flips up and travels along the top of the tube, and in that geometry its spin would then oppose its motion resulting in a loss of energy through friction at the contact point. But I am skeptical that this could be a big enough effect to overcome the "head start" that the early sharp drop provides.

Your experiment is a good case of what happens when the idealized world of physics problem sets --a world of massless pulleys, frictionless planes, and unstretchable strings -- meets the real world. The real world is a LOT more complex than what is taught in most physics courses, since the goal of the courses is to reveal the underlying principles without getting lost in the myriad details of real world situations. This dichotomy becomes painfully apparent the first time a young physicist has to start performing real experiments with actual devices; a problem that takes 15 minutes to solve at the blackboard can take 15 weeks to solve in the lab where the properties of actual things are inescapable.

Your observations are what is important in this case, not our collective inability to come up with an explanation (yet -- maybe another "expert" will figure out what's going on). I am pleased that you persevered with your experiment rather than succumbing to the temptation to ignore or falsely explain your results.

Your experiment is a good case of what happens when the idealized world of physics problem sets --a world of massless pulleys, frictionless planes, and unstretchable strings -- meets the real world. The real world is a LOT more complex than what is taught in most physics courses, since the goal of the courses is to reveal the underlying principles without getting lost in the myriad details of real world situations. This dichotomy becomes painfully apparent the first time a young physicist has to start performing real experiments with actual devices; a problem that takes 15 minutes to solve at the blackboard can take 15 weeks to solve in the lab where the properties of actual things are inescapable.

Your observations are what is important in this case, not our collective inability to come up with an explanation (yet -- maybe another "expert" will figure out what's going on). I am pleased that you persevered with your experiment rather than succumbing to the temptation to ignore or falsely explain your results.

- John Dreher
- Expert
**Posts:**294**Joined:**Sun Dec 25, 2011 8:33 am**Occupation:**Astronomer, Professor of Physics, SETI Researcher (retired)**Project Question:**n/a**Project Due Date:**n/a**Project Status:**Not applicable

Hello Courtney,

John Dreher's reply covers several important issues, but I think the main issue is as follows.

Just the starting height and pipe length are not enough information to determine how long the marble takes to run the course. Just to take an extreme case, suppose the pipe initially gives the marble a steep drop, and then the pipe slopes back up, and finally back down. In other words, your marble roller coaster has a hill in the middle of the run. Now suppose that the hill is adjusted so that the marble nearly stops at the top of the hill. If the marble keeps up speed, the hill may contribute only a little to the total run time, but suppose that the hill is adjusted so that the marble almost comes to a stop. There is no limit to just how slow the marble might get before it starts down the hill again. Tiny changes in the height of the hill would make big changes in the total run time. If it gets slow enough, the total time required to run the pipe might be only a little more than the time required to creep over the hill.

Well, the point is that the time to run the course depends on the exact shape of the pipe, not just the height and overall length.

Calculus is needed to calculate the run time for any complex shape, but I hope this extreme example helps you understand the basic physics.

BTW, even though your marble is inside a pipe, it still encounters air resistance. From the point-of-view of the marble, the air is constantly whizzing by, just as you would feel air in your face when riding a roller-coaster even if there were no wind blowing.

Good luck,

WW

John Dreher's reply covers several important issues, but I think the main issue is as follows.

Just the starting height and pipe length are not enough information to determine how long the marble takes to run the course. Just to take an extreme case, suppose the pipe initially gives the marble a steep drop, and then the pipe slopes back up, and finally back down. In other words, your marble roller coaster has a hill in the middle of the run. Now suppose that the hill is adjusted so that the marble nearly stops at the top of the hill. If the marble keeps up speed, the hill may contribute only a little to the total run time, but suppose that the hill is adjusted so that the marble almost comes to a stop. There is no limit to just how slow the marble might get before it starts down the hill again. Tiny changes in the height of the hill would make big changes in the total run time. If it gets slow enough, the total time required to run the pipe might be only a little more than the time required to creep over the hill.

Well, the point is that the time to run the course depends on the exact shape of the pipe, not just the height and overall length.

Calculus is needed to calculate the run time for any complex shape, but I hope this extreme example helps you understand the basic physics.

BTW, even though your marble is inside a pipe, it still encounters air resistance. From the point-of-view of the marble, the air is constantly whizzing by, just as you would feel air in your face when riding a roller-coaster even if there were no wind blowing.

Good luck,

WW

- wendellwiggins
- Expert
**Posts:**338**Joined:**Sun Jul 10, 2011 5:48 pm**Occupation:**retired physicist**Project Question:**n/a**Project Due Date:**n/a**Project Status:**Not applicable

Wendell is, of course, right. But the puzzle, from my perspective, is that a run with an early quick drop, like your case B, ought to be faster than one without an early drop, like your case A. I can't prove it (without a lot of effort), but my guess is that for idealized (completely rigid)pipe the fastest path would be a vertical drop followed by a horizontal run -- that way you maximize the speed as soon as possible. Wendell has pointed out that you can make the track as slow as you want with an appropriately shaped hump. Alas, doing this problem right, with real dynamics and friction and deformable pipe would be very difficult.

- John Dreher
- Expert
**Posts:**294**Joined:**Sun Dec 25, 2011 8:33 am**Occupation:**Astronomer, Professor of Physics, SETI Researcher (retired)**Project Question:**n/a**Project Due Date:**n/a**Project Status:**Not applicable

Thanks Wendell and John. What I hear you both saying is that just because the marble has the identical potential energy at the beginning of both case A and case B does not mean it must run the tubing in the same amount of time. Got it. It seems so obvious now, when one considers the example Wendell gives where the second hill becomes much taller....you can just see the marble losing speed as it struggles to reach the top. So...got it, just because the length of the pipe is the same, and the friction the same, and the air resistance--even in the tube--the same, doesn't mean the conversion of potential energy to kinetic energy will take the same amount of time.

So now what?

What we need to do is retroactively create a bit more challenge to this science fair project where the initial question and hypothesis seem, in retrospect, way too simplistic for a 5th grader.

Two ideas come to mind:

1) Can we use 5th grade math to make a basic prediction about the marble's speed along the 3 straight slope courses (A, C, D)? I see that there is some discussion about calculating the velocity of an item on a slope ( https://www.google.com/search?q=calcula ... =firefox-a ). We could make a "retroactive" prediction about the marble's time to run the course based on the differential of the slope.

2) Even though the marble in case A gets through the course faster is it actually traveling faster at the point of exit compared to the marble in course B? We could use a gram scale held vertically to measure the force of the marble as it exits course A and course B (force being a proxy for the actual speed since we don't have an easy way to measure that since we can't see the marble reach the top of the second hill). In theory, since the marble in course B had to "work" harder to reach the top of the second hill, that work or energy used to get up the second hill will be stored in the marble at the top of the second hill. As the marble then runs down the second hill, it will be going faster than the marble in course A at the same point. If we go this route, are we just proving the obvious again? Or is there enough nuance in this question?

Thanks,

Courtney

So now what?

What we need to do is retroactively create a bit more challenge to this science fair project where the initial question and hypothesis seem, in retrospect, way too simplistic for a 5th grader.

Two ideas come to mind:

1) Can we use 5th grade math to make a basic prediction about the marble's speed along the 3 straight slope courses (A, C, D)? I see that there is some discussion about calculating the velocity of an item on a slope ( https://www.google.com/search?q=calcula ... =firefox-a ). We could make a "retroactive" prediction about the marble's time to run the course based on the differential of the slope.

2) Even though the marble in case A gets through the course faster is it actually traveling faster at the point of exit compared to the marble in course B? We could use a gram scale held vertically to measure the force of the marble as it exits course A and course B (force being a proxy for the actual speed since we don't have an easy way to measure that since we can't see the marble reach the top of the second hill). In theory, since the marble in course B had to "work" harder to reach the top of the second hill, that work or energy used to get up the second hill will be stored in the marble at the top of the second hill. As the marble then runs down the second hill, it will be going faster than the marble in course A at the same point. If we go this route, are we just proving the obvious again? Or is there enough nuance in this question?

Thanks,

Courtney

- Courtney
**Posts:**42**Joined:**Thu Sep 18, 2003 5:36 pm

P.S.

Now I am wondering, maybe the question of the speed of A and B at exit is not as obvious as I thought. Now I am even wondering which marble would be going faster at the exit point, A or B? Maybe the marble in A actually is going faster because it is accelerating the entire length of the pipe whereas B is accelerating down a steeper slope but has just begun to accelerate. Maybe this is not simple enough at all. When Matt gets home from school, I am going to ask him what he thinks. But I am thinking maybe we would be back to calculus again in trying to predict the outcome of this question. Would it be possible to explain, with 5th grade math, why one marble is going faster than the other at exit?

Now I am wondering, maybe the question of the speed of A and B at exit is not as obvious as I thought. Now I am even wondering which marble would be going faster at the exit point, A or B? Maybe the marble in A actually is going faster because it is accelerating the entire length of the pipe whereas B is accelerating down a steeper slope but has just begun to accelerate. Maybe this is not simple enough at all. When Matt gets home from school, I am going to ask him what he thinks. But I am thinking maybe we would be back to calculus again in trying to predict the outcome of this question. Would it be possible to explain, with 5th grade math, why one marble is going faster than the other at exit?

- Courtney
**Posts:**42**Joined:**Thu Sep 18, 2003 5:36 pm

I'm not sure how one could use just 5th grade math to predict the results; however, some fifth graders are capable of doing math beyond the 5th grade level. Understanding a simple algebraic equation or formula and how to variable substitution (substitute the measurement numbers from the experiment into the appropriate equation variables) is definitely needed. Simple fractions like 1/2 and squaring a number (number times itself) are also be required if you want to involve distance as well as speed.

The basic physics concept involves a rate of change equation (the derivation of which requires calculas); however, the derived formulas can be used (without understanding their derivation). As a marble rolls down a constant slope hill, it keeps accelerating (keeps gaining speed at a constant rate). As a marble rolls up a constant slope hill, it keeps de-accerating (keeps loosing speed at a constant rate). Whether this is beyond your 5th grader's ability to understand might depend on your ability to find materials to explain it in a way that your son can understand. There are different ways to approach these problems and use analogies to your son's previous experiences.

The formulas for curved secions involve a rate of change in a rate of change equation which usually defies simplification. Engineers often use approximations for these cases. If you approximate a curved section with several straight sections, you can come close to the answer. One of the fundamental basis for calculas comes from a Limit Theory: If you keep breaking something up into more simple pieces (where the math is easy) and you sum up the results for all the pieces, you get closer and closer to the mathmatical answer.

The basic physics concept involves a rate of change equation (the derivation of which requires calculas); however, the derived formulas can be used (without understanding their derivation). As a marble rolls down a constant slope hill, it keeps accelerating (keeps gaining speed at a constant rate). As a marble rolls up a constant slope hill, it keeps de-accerating (keeps loosing speed at a constant rate). Whether this is beyond your 5th grader's ability to understand might depend on your ability to find materials to explain it in a way that your son can understand. There are different ways to approach these problems and use analogies to your son's previous experiences.

The formulas for curved secions involve a rate of change in a rate of change equation which usually defies simplification. Engineers often use approximations for these cases. If you approximate a curved section with several straight sections, you can come close to the answer. One of the fundamental basis for calculas comes from a Limit Theory: If you keep breaking something up into more simple pieces (where the math is easy) and you sum up the results for all the pieces, you get closer and closer to the mathmatical answer.

-Craig

- Craig_Bridge
- Expert
**Posts:**1297**Joined:**Mon Oct 16, 2006 11:47 am

You asked whether you can use 5th grade math to make basic predictions about the marble's speed along the 3 straight slope courses (A, C, D).

Yes. The three cases have different starting heights and, therefore, different amounts of potential energy. At the base of each run the potential energy will have been converted to kinetic energy. The kinetic energy is proportional to speed squared, and the potential energy is proportional to height. If you graph v squared against h you should get a straight line. The hard part would be to measure the speed at exit. Your idea of using a scale is ingenious if you can pull it off. In fact it adds more physics, since the scale deflection converts the kinetic energy back into potential energy, in this case the potential energy of the compressed spring, which is proportional to the displacement squared, so the energy is 1) the potential energy due to gravity, A*h where A is a constant, and h is the starting height; 2) the kinetic energy of motion at the exit point, B*v² where B is a constant, and v is the velocity; and 3) the potential energy of the compressed spring, C*d² where C is a constant, and d is the displacement of the spring; conservation of energy then means that all three of these energies must be the same (less small losses due to friction). A plot of d² versus H should be a straight line. Now you may be concerned that friction is not small. If you were sliding blocks down the tubes, friction would not be small. But you are rolling marbles. and rolling gets rid of almost all the friction — that’s why the wheel was such an important invention. The rolling motion does add a little complexity to computing the kinetic energy, it is no longer just (½)*m*v², but it is still proportional to v² (and to m). (In fact, if my memory serves me it is [(½) + (1/5)]*m*v² where the second term accounts for the kinetic energy of rotation of the marble.) Bottom line: if you can figure out how to capture the spring deflection quantitatively, then you can make a plot showing conservation of energy without anything beyond understanding formulae such as C*d² and graphs. This method can be extended to tracks of any shape — this is the power of conservation laws.

On the other hand, if you wish to compute the time it will take for the marbles to run the three straight courses A, B, and D, you will need math and physics well beyond the 5th grade level. You would need to start with Newton’s 2nd law, F=m*(dv/dt) where the term (dv/dt) is the derivative of the velocity as a function of time; this introduces calculus at the very first step. Next you would need to analyze the forces involved, which would involve simple trigonometry. Finally you would need to understand the basics of rotation of a rigid body, which most college freshmen find a bit challenging. Bottom line: a “no go” for grade 5.

Yes. The three cases have different starting heights and, therefore, different amounts of potential energy. At the base of each run the potential energy will have been converted to kinetic energy. The kinetic energy is proportional to speed squared, and the potential energy is proportional to height. If you graph v squared against h you should get a straight line. The hard part would be to measure the speed at exit. Your idea of using a scale is ingenious if you can pull it off. In fact it adds more physics, since the scale deflection converts the kinetic energy back into potential energy, in this case the potential energy of the compressed spring, which is proportional to the displacement squared, so the energy is 1) the potential energy due to gravity, A*h where A is a constant, and h is the starting height; 2) the kinetic energy of motion at the exit point, B*v² where B is a constant, and v is the velocity; and 3) the potential energy of the compressed spring, C*d² where C is a constant, and d is the displacement of the spring; conservation of energy then means that all three of these energies must be the same (less small losses due to friction). A plot of d² versus H should be a straight line. Now you may be concerned that friction is not small. If you were sliding blocks down the tubes, friction would not be small. But you are rolling marbles. and rolling gets rid of almost all the friction — that’s why the wheel was such an important invention. The rolling motion does add a little complexity to computing the kinetic energy, it is no longer just (½)*m*v², but it is still proportional to v² (and to m). (In fact, if my memory serves me it is [(½) + (1/5)]*m*v² where the second term accounts for the kinetic energy of rotation of the marble.) Bottom line: if you can figure out how to capture the spring deflection quantitatively, then you can make a plot showing conservation of energy without anything beyond understanding formulae such as C*d² and graphs. This method can be extended to tracks of any shape — this is the power of conservation laws.

On the other hand, if you wish to compute the time it will take for the marbles to run the three straight courses A, B, and D, you will need math and physics well beyond the 5th grade level. You would need to start with Newton’s 2nd law, F=m*(dv/dt) where the term (dv/dt) is the derivative of the velocity as a function of time; this introduces calculus at the very first step. Next you would need to analyze the forces involved, which would involve simple trigonometry. Finally you would need to understand the basics of rotation of a rigid body, which most college freshmen find a bit challenging. Bottom line: a “no go” for grade 5.

- John Dreher
- Expert
**Posts:**294**Joined:**Sun Dec 25, 2011 8:33 am**Occupation:**Astronomer, Professor of Physics, SETI Researcher (retired)**Project Question:**n/a**Project Due Date:**n/a**Project Status:**Not applicable

First off, thanks again to all of you for putting so much thought into this for us. We probably should have posted this question to the K-5 forum, but I somehow felt the questions we were were asking went beyond that level...and they were, as we can all probably agree. Matt's great at math, but he's nowhere near understanding these kinds of equations. What we decided to do is see if Matt could predict the time it would take the marble to run the course at two additional starting heights (straight course, no curves), based on graphing the run times and starting heights of A, C, and D. So he plotted those points on a line on graph paper and then predicted the run times for the new courses (by looking at where the points for the new course heights should fit on the line of run times). He then tested his predictions by running the two new courses. In short, it worked as you might expect. No college math or even middle school math, but the experiment still serves as a focal point for him having learned first hand about conservation of energy, slope (rise:run), acceleration of gravity, velocity, gravity, and friction. I think he'll go ahead and calculate the velocity of the marbles, too, in miles per hour (just to put the speed into familiar terms). For fun, we tested using the gram scale to record the force of the marble at exit -- that was pretty cool. It's only a $25 digital kitchen scale, but it was pretty easy to detect the differences in the force of the marble at different speeds as measured in grams. It really illustrates how you are not measuring mass anymore...but rather ma (F = ma). He's sleeping now, but tomorrow morning I am going to ask him what, in terms of conservation of energy, a marble running down a PVC pipe has in common with the bullets traveling down the gun barrels in his favorite video game and the pneumatic mechanism in the paint ball guns he was firing at a friend's recent birthday party. The better he can articulate this, the more he can make up for the relative simplicity of his actual experiment by demonstrating his understanding of the underlying concepts to the judges. Thanks again, gentlemen!

- Courtney
**Posts:**42**Joined:**Thu Sep 18, 2003 5:36 pm

Our pleasure.

BTW, conservation of energy is by no means as "easy" as it might appear. I still wonder: "How does the universe keep the books on conservation of energy, how can an elementary particle like an electron know that it has so much potential energy and so much kinetic energy and so much rest mass energy?"

BTW, conservation of energy is by no means as "easy" as it might appear. I still wonder: "How does the universe keep the books on conservation of energy, how can an elementary particle like an electron know that it has so much potential energy and so much kinetic energy and so much rest mass energy?"

- John Dreher
- Expert
**Posts:**294**Joined:**Sun Dec 25, 2011 8:33 am**Occupation:**Astronomer, Professor of Physics, SETI Researcher (retired)**Project Question:**n/a**Project Due Date:**n/a**Project Status:**Not applicable

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