Rolling ball down ramp to calculate acceleration of gravity

@puzzledperson,

Oh, so NOW you want us to ignore... the existence of [earth's] atmosphere?

It wasn't ignored. It was there in each case.

But it is irrelevant it any event, unless you're talking about a feather or something, or perhaps where you're talking about extremely long falls where terminal velocity can be reached. But the terminal velocity (regarding gravitational force) of a bullet which is dropped or shot from a rifle would be the same any way, so it wouldn't change the outcome.

1 Reply

@maxdancona,

maxdancona wrote: "If you want to make this argument, we can calculate the orbital velocity. Then we can take some amount off of this and calculate the path the object will take and see where it will intercept the Earth."

Ignoring air resistance and intervening objects, if you are allowed to specify muzzle velocities which are arbitrarily close to, but less than, the orbital velocity, you can make a projectile travel around the Earth an arbitrarily large number of times, though it will slowly sink toward the Earth's surface until it hits. Therefore it will take an arbitrarily long time to stop circling the Earth. That makes perfect sense. The difference between us is that I can perceive that using mathematical inference, and you can't.

maxdancona wrote: "(Actually the "orbital velocity" is an strange term. again, if we are going to discuss this we should define this mathematically so that we can be exact and get the correct answer)."

The orbital velocity is the muzzle velocity at which the forward movement of the projectile offsets the downward movement caused by gravity so that the distance of the projectile from the Earth remains constant. Nothing strange about the term or its usage. Of course, you have to ignore air resistance, intervening objects, and perturbational phenomena which affect the Earth differently from the projectile, as well as nongravitational forces.

maxdancona wrote: "If you are not willing to do the math, then you are just making stuff up."

What you don't realize is that mathematical reasoning isn't limited to deductive proofs involving equations. It can also be inductive and inferential.

1 Reply

@puzzledperson,

Quote:

What you don't realize is that mathematical reasoning isn't limited to deductive proofs involving equations. It can also be inductive and inferential.

That aint mathematical reasoning, it's just plain, everyday logic.

Math is a subset of logic. "Applied logic," ya might say. Lots of ways to apply logic. Math aint the onliest way.

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@layman,

Of course air resistance would change the outcome, because we're talking about nonlinear effects. Air resistance is a negligible factor when calculating the time it takes a lead weight (bullet) to hit the ground when dropped from shoulder height. But for a bullet fired from a rifle at from high (and varying) velocity, air resistance is a huge factor in determining flight time.

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@layman,

Max thinks math IS logic. And he also thinks (wrongly) that physics is math. He just can't make distinctions.

Euclidean geometry is another system of applied logic. It's basic axioms and theorems have nothing to do with math, per se.

Example: Logic will tell you that the area of a triangle will be 1/2 its base times it height. Now if you want to know HOW MUCH area is in a particular triangle, you can haul out your slide rule and calculate it. But that will NEVER tell you WHY the formula works. Logic tells you that, not math.

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@layman,

Quote:

Example: Logic will tell you that the area of a triangle will be 1/2 its base times it height. Now if you want to know HOW MUCH area is in a particular triangle, you can haul out your slide rule and calculate it. But that will NEVER tell you WHY the formula works. Logic tells you that, not math.

(Actually logic is a branch of mathematics... but I will play along.)

Please tell me why the formula works without using math.

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@puzzledperson,

Quote:

But for a bullet fired a rifle at from high (and varying) velocity, air resistance is a huge factor in determining flight time.

How, and in what way? Are you talking about projectiles fired at DIFFERENT angles, e. g., 30 degrees or 45 degrees? I was talking about a rifle fired at a 90 degree angle (from the center of gravity).

1 Reply

@layman,

layman wrote: "Math is a subset of logic."

Didn't Bertrand Russell attempt to prove that?

Knowing that X+1 always yields a larger number, for any natural number, allows me to conclude, using mathematical induction, that there is no largest natural number. That IS a type of mathematical reasoning. But it isn't a deductive proof based on equations involving specific numbers: and in fact such a proof is impossible, for the simple reason that the unlimited quantity of potential natural numbers would require a deductive proof that could never be completed.

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@maxdancona,

Quote:

(Actually logic is a branch of mathematics... but I will play along.)

Ummm, actually, no. Logic is NOT a branch of mathematics. Do some research if you don't know that. Good thing you're "playing along," though.

Quote:

Please tell me why the formula works without using math.

I didn't say the FORMULA works without using math. Read it again. Who knows, there's a slight possibility you might comprehend the point, if you really try (but at this point, I doubt it).

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@puzzledperson,

Quote:

That IS a type of mathematical reasoning.

I disagree. That is just reasoning which, in this case, happens to be ABOUT math. But it is not "mathematical reasoning."

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@layman,

Layman wrote:

Now if you want to know HOW MUCH area is in a particular triangle, you can haul out your slide rule and calculate it. But that will NEVER tell you WHY the formula works. Logic tells you that, not math.

You said that you can, without using math, tell me WHY the formula works.

Please do that.

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@maxdancona,

Quote:

Please tell me why the formula works without using math.

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.

1 Reply

@layman,

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.

1 Reply

@layman,

No, you can limit it to a rifle fired parallel to the ground.

If air resistance is negligible for a lead weight dropped by hand from shoulder height, then its drop is governed by gravity alone and its acceleration is linear, that is, 32 feet per second, per second. 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. Finally, drag and lift are affected by the shape and mass distribution of the bullet, as well as the rifling which affects how it revolves, so that different bullets fired with the same muzzle velocity may behave differently, even when they weigh the same.

Of course, in the absence of air resistance, the bullet's muzzle velocity would remain its constant forward velocity until it hit something (like the ground).

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@layman,

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?

https://encrypted-tbn1.gstatic.com/images?q=tbn:ANd9GcR6iVis7JIGwj1g4Yd06y_4WtFeRj4KYYO7d4GhnYYw0TfUKiLDdw

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@maxdancona,

Quote:

What rectangle to you cut in half to get this triangle

Max, are you ever capable of getting a GENERAL point, or can you only understand particular instantiations of it?

Cut a circle in half, and each half will have equal areas.

Cut a parallelogram in half, and each half will have equal areas.

See the GENERAL, LOGICAL point? Or not?

1 Reply

@layman,

But the triangle I gave you is not a triangle you get from cutting a rectangle in half (any such triangle will have one 90 degree angle).

I can still prove to you that this triangle, or any other triangle you can possibly give me, will follow the rule... Area = 1/2 base * height.

This is the power of mathematics. I can show you the mathematical proof if you would like... Actually everyone who takes a high school geometry class does this proof for themselves.

1 Reply

@maxdancona,

Quote:

This is the power of mathematics.

Wrong. That is the power of logic, not math.

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@puzzledperson,

No. 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.

Well, you've given your opinion. You've accepted your opinion as fact.

1 Reply

@layman,

Quote:

Ummm, actually, no. Logic is NOT a branch of mathematics. Do some research if you don't know that. Good thing you're "playing along," though.

Maybe this math professor can help you, eh, Max?:

Quote:

Logic is the science of formal principles of reasoning or correct inference...What then is reasoning? According to Aristotle [13, Topics, 100a25], reasoning is any argument in which certain assumptions or premises are laid down and then something other than these necessarily follows. Thus logic is the science of necessary inference....

However, when logic is applied to specific subject matter...any decisions about what is true and primary do not pertain to logic but rather to the specific subject matter under consideration. In this way we limit the scope of logic, maintaining a sharp distinction between logic and the other sciences.

Mathematics is the science of quantity....The purpose of this section is to indicate the role of logic in the foundations of mathematics.

http://www.personal.psu.edu/t20/papers/philmath/

If this isn't enough to show you that math is just applied logic (applied to a particular (DISTINCT) science--the science of quantities), then do some of your own research. The difference is certainly explained better elsewhere--this is just the first site I clicked on.

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Rolling ball down ramp to calculate acceleration of gravity

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