Rolling ball down ramp to calculate acceleration of gravity

@puzzledperson,

Quote:

If it is fired at high velocity from a rifle, then drag and lift become relevant as well as gravity. Furthermore, they will decrease the speed of the bullet nonlinearly. Of course, in the absence of air resistance, the bullet's muzzle velocity would remain its constant forward velocity.

As I've said a few times already. My example was NOT about velocity. It was designed to show that velocity is not the issue. The question of how FAR it travels in a given time is distinct from the question of how far it falls in a given time.

If you have other forces in play, such as a strong wind upwards or downwards, then of course things change. That has nothing to do with gravity.

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@maxdancona,

maxdancona wrote:

First of all that is a mathematical argument... but second of all, it is missing something.

What rectangle do you cut in half to get this triangle?

The article you posted is very interesting. Did you read it? I hope you didn't just skim through it looking for snippets that seemed to say what you wanted them to? I don't like your habit of arguing by Google search, but I must admit that once in a while you come up with some good articles. I hope that you read and try to understand these articles in their entirety rather than just looking for one liners. The relationship between Platoism and predicate calculus would be an interesting discussion.

But back to the interesting challenge that you, yourself, suggested. You claimed that you could use logic to prove that the area of a triangle is equal to the one half the base times the height.

This is very instructive to our discussion.

Please do what you said you could do. I have provided the triangle.

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@maxdancona,

I'm not interested in doin geometrical problems here, Max. I was using geometry to make a point. Do you understand the point? Logic is not math, and math is not logic. Math uses logic, just like physics uses math (and logic).

Euclidean geometry starts with definitions, axioms, etc. Now, the question is, if you ASSUME the "truth" of those axioms, what necessarily follows by logical implication? Euclidean geometry says nothing about the world. It may or may not be "true" in that sense. In fact Einstein had to turn to non-Euclidean geometry in GR to develop his theory.

Logical necessity is not a question about math. It is a question about logic. I have used contrasting syllogisms in the other thread to show that "validity" is a question which says nothing about the "soundness" of the argument (the "truth" of the premises).

You can't understand anything I say because you think that physics is math, and that logic is math, etc. You have no bearings. Everything is math to you.

Quote:

Logic is the science of formal principles of reasoning or correct inference... we limit the scope of logic, maintaining a sharp distinction between logic and the other sciences.

Mathematics is the science of quantity.

That's the point of citing the article. Get it yet?

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@maxdancona,

maxdancona wrote:

maxdancona wrote:

First of all that is a mathematical argument... but second of all, it is missing something.

What rectangle do you cut in half to get this triangle?

To briefly respond to this "puzzle:" I aint no geometrician, but I think what you need to do here, logically, not mathematically, is find the point on the line BC, where you can draw a line from A to that point which gives you TWO right-angled triangles. Then each of those can be reduced to the 1/2 base times height formula, and you can calculate the area from there IF that's your purpose. You don't have to know, or even calculate that QUANTITY (math is "the science of quantities," remember?) in order to use logic to understand WHY the formula would work. Logic, NOT math, tells you that.

1 Reply

@maxdancona,

Quote:

Logic is the science of formal principles of reasoning or correct inference

Mathematics is the science of quantity.

I'm crossing threads here to some extent, but even in this thread you have flatly (but quite erroneously) stated that "physics is math." That too demonstrates a fundamental misunderstanding and it helps explain why you can't comprehend a single point I made in the "special relativity" thread. You continue to make this assertion EVEN AFTER Feynman tells you you're wrong about it.

As a simple matter, it has been said that physics is the "science of matter in motion." That is NOT math. That is NOT logic, either. That is physics, and there's a difference.

Physics merely USES math and logic as tools which aid it's study of matter in motion.

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layman wrote:

Quote:
OK, let me explain it a little. If you cut one thing into two equal parts the two parts will be equal. Get it? That aint math. It's just simple logic.

Maybe I need to take it one step further for you, I don't know:

Cut a rectangle in half, and the area of each half will be one half of the total for the rectangle.

What's up with that triangle. Layman said "Cut a rectangle in half, and the area of each half will be one half of the total for the rectangle." He didn't say a word about cutting a rectangle to produce a triangle.

1 Reply

@Kolyo,

Kolyo wrote: "You just need to analyze the situation in terms of gravitational potential energy, rotational kinetic energy, and translational kinetic energy."

The problem is that gravitational potential energy depends solely on the height of the ramp, not the length of the ramp.

If the ramp is 10 feet long, but supported by table legs that make it 3 feet tall (as measured from the top of the ramp straight down to the floor), then the potential energy is that of a ball suspended 3 feet above the ground.

If the ramp is 10 feet long, but supported by a pair of dimes, so that it is almost but not quite parallel to the floor, then the potential energy is that of a ball suspended from a height equal to the thickness of a dime (plus the thickness of the ramp).

But the length of the ramp is the same in each case. So quite different amounts of potential energy will be expended over an identical distance. When a smaller amount of energy is expended over the same distance, the acceleration will be less and the time of the ball's journey down the length of the ramp will be longer.

This is true even if friction is negligible and the ball slides down the ramp without rolling.

The original question asked why the acceleration didn't match that of gravity. The only way it could would be if the board were standing vertically instead of inclined as a ramp. Then the length of the board would also be the height of the ball off the ground. Every inclination of the ramp less than 90 degrees will give a ball at the top less potential energy.

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@roger,

roger wrote: "The time it takes your bullet to hit the ground has nothing to do with its velocity, and air resistance is only a factor for the six feet between the rifle and the ground."

If you drop a frisbee from shoulder height, gravity is the only force affecting it. If you throw the frisbee properly, aerodynamic forces of lift and drag also come into play, and the time it takes to hit the ground is not the same.

It might seem counterintuitive that a bullet could be affected by aerodynamic forces, but when a bullet is traveling very fast the air is in effect rushing over the bullet very fast, rather like a wind-tunnel.

"The combined sideways wind component of these two effects causes a Magnus force to act on the bullet, which is perpendicular both to the direction the bullet is pointing and the combined sideways wind. In a very simple case where we ignore various complicating factors, the Magnus force from the crosswind would cause an upward or downward force to act on the spinning bullet (depending on the left or right wind and rotation), causing an observable deflection in the bullet's flight path up or down, thus changing the point of impact."

https://en.m.wikipedia.org/wiki/Magnus_effect

The problem is complicated by such things as a bullet's shape. Modern bullets are not uniform spheres, they are sharp or blunt cylindrical projectiles, with different distributions of mass, different centers of gravity, and different balances along both longitudinal and lateral axes. If a bullet passing very quickly through the air gets tilted slightly upward or downward, aerodynamic forces may push down or up on it, thus affecting flight time.

1 Reply

@roger,

Quote:

What's up with that triangle. Layman said "Cut a rectangle in half, and the area of each half will be one half of the total for the rectangle." He didn't say a word about cutting a rectangle to produce a triangle.

No, he didn't say anything about it, but I did. If you cut a rectangle in half diagonally you will have two equal triangles instead of just one rectangle. Hence the formula for the area is one-half of base times height instead of just base times height, which is the formula for the whole rectangle.

The point was that you don't need math to deduce that, just logic.

1 Reply

@sjlcomp,

You wrote in your initial question that the measured path of the ball was 3 feet, and that the angle of the ramp with the floor was 27 degrees and that it took 0.9 seconds.

If we consider the full length of the ramp to be the same (3 feet), then we can view the ramp as the hypotenuse of a right triangle with the vertical rise from the floor to the top of the ramp to be determined by the formula:

Height = (sin 27 degrees) * 3 ft = 1.362 ft.

The height of the ramp is .454 of the length of the ramp, so I would expect the acceleration of the ball down the 3 feet of ramp to be slightly less than half that of gravity, notwithstanding losses to friction and to rotational momentum.

Rate = Distance / Time = 3 / 0.9 = 3.333333 feet per second. But since that is the average rate and the initial rate was zero, and acceleration is linear, the terminal velocity is twice that, or 6.666666 feet per second.

At the end of 1 second, acceleration should equal terminal velocity since the ball began at rest at T = 0 and acceleration is the rate of change of velocity PER SECOND.

6.666666 / 32 = .2083 which is substantially less than the .454 ratio I described above. However, there is loss of energy to friction as well as conversion of potential energy into rotational rather than translational energy to be accounted for. There is also the possibility of various kinds of measuring error.

1 Reply

@layman,

layman wrote:

If you cut a rectangle in half diagonally you will have two equal triangles instead of just one rectangle. Hence the formula for the area is one-half of base times height instead of just base times height, which is the formula for the whole rectangle.

The point was that you don't need math to deduce that, just logic.

And Max's point (I think, since I've just been skimming) was that your argument doesn't work as a valid proof that bh/2 describes the area of ANY triangle. You've merely shown that bh/2 is the correct area formula for a right triangle. It can (of course) be shown that the area of any triangle is bh/2. But you haven't done it.

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@Kolyo,

The other thing that makes me laugh is the idea that A = 1/2 bh is not math.

Even the statement that cutting a rectangle into two equal pieces results in pieces that are 1/2 the area of the original rectangle is math.

I don't accept Aristotle's statement that "math is the science of quantity". I don't think most modern mathematicians accept that either.

But even if I did, 1/2 is a quantity.

At any rate, this thread is silly. For anyone who wants the real answer... the main error in the original poster's calculations was ignoring the term for the energy in the angular momentum of the rolling ball.

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@layman,

layman wrote:

To briefly respond to this "puzzle:" I aint no geometrician, but I think what you need to do here, logically, not mathematically, is find the point on the line BC, where you can draw a line from A to that point which gives you TWO right-angled triangles. Then each of those can be reduced to the 1/2 base times height formula, and you can calculate the area from there IF that's your purpose. You don't have to know, or even calculate that QUANTITY (math is "the science of quantities," remember?) in order to use logic to understand WHY the formula would work. Logic, NOT math, tells you that.

Yep, that's what you'd do.

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@maxdancona,

Yeah kind strange that division doesn't count as math.

1 Reply

@Kolyo,

Quote:

And Max's point (I think, since I've just been skimming) was that your argument doesn't work as a valid proof that bh/2 describes the area of ANY triangle. You've merely shown that bh/2 is the correct area formula for a right triangle.

That wasn't my argument, and that wasn't my point. The point was about logic versus math, not geometry per se.

I didn't say ANY triangle. Max just tried to make it into that.

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@maxdancona,

maxdancona wrote:

The other thing that makes me laugh is the idea that A = 1/2 bh is not math.

Even the statement that cutting a rectangle into two equal pieces results in pieces that are 1/2 the area of the original rectangle is math.

I don't accept Aristotle's statement that "math is the science of quantity". I don't think most modern mathematicians accept that either.

But even if I did, 1/2 is a quantity.

A dog is math to you, Max.
The kitchen floor is math to you.
A 3 foot chunk of lead pipe upside your head is math to you.

EVERYTHING is math to you.

What else is new? That's why you'll never grasp any real nuances or distinctions about math, physics, logic, or virtually anything else. All you can see, whatever the topic, is one big number in front of you

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@Kolyo,

Kolyo wrote:

Yeah kind strange that division doesn't count as math.

Quote:

Max said: The other thing that makes me laugh is the idea that A = 1/2 bh is not math

You're both missing the point. Apparently you just skimmed the post in question.

This AINT the point, but just to respond: I can take out a samuri sword and chop a pumpkin in half without any math. But you would probably call choppin "division" and try to turn the word "half" into .50 so you could call it "math."

I was talking about how you deduce and explain the applicability (not the application) of the formula.

I wasn't even talking about APPLYING the formula---you can call that math, if you want, but that wasn't the point

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Goin back to our trusty Penn State math professor again, this was the point:

Quote:

This attitude about logic is in agreement with the modern view, according to which the predicate calculus (see 1.2 below) is a general method or framework not only for philosophical reasoning but also for reasoning about any subject matter whatsoever.

Max said logic was "a branch of mathematics." Fraid not, eh?

You don't need any math to apply logic, and it can be applied to any topic, including math. Furthermore, just because logic is USED to establish mathematical concepts and to deduce conclusions in the realm of math, that doesn't make logic math. Without logic there would be no math. Math is STRICTLY about drawing correct inferences from pre-established premises.

Math is just "applied logic."

1 Reply

@puzzledperson,

I wrote: "At the end of 1 second, acceleration should equal terminal velocity since the ball began at rest at T = 0 and acceleration is the rate of change of velocity PER SECOND."

This is true, but 6.666666 is the terminal velocity after 0.9 seconds, not 1 second. So the acceleration would be:

6.666666 / 0.9 = 7.41 feet per second per second.

So the acceleration is 0.23 that of gravity.

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@layman,

I think layman is correct in asserting that math is a subset of logic. For example, arithmetic is a set of deductive rules based on premises defining relations between symbols ("numbers"), as well as certain other premises (e.g. the law of the excluded middle, etc.).

To multiply two numbers, one can either work from established tables, or carry out a series of repeated additions defined to be equivalent to multiplication. More complicated products are obtained from composite groups, or concatenations of such processes.

Even something as complicated as calculus is based on a set of rules governing relations between elements.

The fact that such elements are interpreted as numeric quantities is irrelevant. Mathematics is a formal system, or set of formal systems, which is why it can be implemented by computers using electronic components designed to be isomorphic to Boolean logic elements. (Though in practice finite precision constraints may result in the use of difference equations rather than differential equations, for example.)

Which is not to say that considerable creativity cannot enter into mathematics. It does, through the selection of premises and rules defining relations and processes, as well as in the adaptation of mathematics to applications.

But beyond that I would have to say that logic is a tool of reasoning, not the whole of it. Because all logical proofs and arguments are finite (else their conclusions could not be reached), they necessarily begin with premises which are stipulated axiomatically. While those axioms might in turn be proven as the conclusions of a second argument, that second argument will itself necessarily rely on stipulated premises. So the need to base logical arguments and proofs on stipulated premises cannot be escaped.

Premises can be adopted because they are directly perceived to be necessary truths (or because they are believed to be or assumed to be such). They can also be adopted as provisional or working premises. And finally, they can be entirely arbitrary.

The selection of rules or methods of reasoning to be accepted as valid, is also in that sense a metalogical process.

1 Reply

Rolling ball down ramp to calculate acceleration of gravity

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