Why is there no dust on the Lunar Lander's footpads?

[Valis]

Actually, there is a limit. An object dropped from an infinite distance would (disregarding the infinite time taken to fall) hit the surface at surface escape velocity. This is quite different from the concept of a terminal velocity, however.

True. My use of the words "without a limit" above is misleading, what I rather meant was that there isn't a "cut-off distance" in the sense Zakalwe suggested, and if you'd increase the mass of the Moon, you'd also increase the impact velocity.

[Chew]

However, the higher object's impact velocity is  v₀+v.

That ain't right.

Consider two objects dropped on the Moon, one at 11 km, the other at 10 km. As the higher object passes the lower object it will be traveling at 57 m/s. But the objects will hit at 189 and 180 m/s.


See the 7th and 8th equations at this link for the equations used where the fall distance is significant to the radius of the Moon: Equations for a falling body

[Valis]

That ain't right.

Consider two objects dropped on the Moon, one at 11 km, the other at 10 km. As the higher object passes the lower object it will be traveling at 57 m/s. But the objects will hit at 189 and 180 m/s.


See the 7th and 8th equations at this link for the equations used where the fall distance is significant to the radius of the Moon: Equations for a falling body

Again, correct (trying to keep my kid happy while I'm typing, not really good for any coherent thought...). Of course the initial velocity will lower the time the gravitational acceleration has for acting on the falling object, resulting in a smaller difference at impact.

[JayUtah]

All good examples and demonstrations.  Not to put too many words in Gillianren's mouth, because her words are so much better than mine, but the sticky wicket for most non-mathematicians and non-physicists is the notion that an abstract arithmetic operation has meaning to a real-world object such as a unit of time.  You can't square a second anymore than you can take the square root of a gallon.  Of course you can in engineering physics, but what does that mean in real-world terms?  Not even engineers typically go there.

The confusion arises from the dual nature of specific measurements as practical aids to daily life -- "Add a teaspoon of vinegar" -- and also as representatives of the underlying quantities and values that play in our investigation of the behavior of the physical world -- "The rate of decay is slowing."  The latter requires reasoning about quantities, rates, and combinations using increasingly more abstract relationships and thus requiring increasingly more formal notation and techniques that end up squaring seconds, square-rooting gallons, "arbitrarily" halving the mass, or similarly confusing calculus fu.  Units jump the shark at that point, when considered as the practical measurements they are to most of us.

[gillianren]

No, that's exactly what I meant.  It's like in . . . Swiftly Tilting Planet, I believe, where Mr. Murry and Charles Wallace are building a model of a tesseract.  It's an abstract mathematical concept (and made The Avengers a little weird for me), and I wasn't sure how you would build a model of one any more than I can get a feel for "seconds squared."

[nomuse]

Well, I've seen 3d models of a 4d hypercube.  Sorta like the usual 2d representation of a 3d wireframe cube.  Except, of course, I don't think I've ever seen one of those models in the flesh, so I've only seen a 2d representation or image of a 3d model of a 4d concept...

Anyhow, I sorta figured that's what they meant.  Although that sentence bugged me.  (Not as much as a later one, in the second book...about Charles Wallace just "needing to adapt...!")

[ka9q]

the sticky wicket for most non-mathematicians and non-physicists is the notion that an abstract arithmetic operation has meaning to a real-world object such as a unit of time.

The really interesting thing is how often you can find a real-world meaning to the units on some set of measurements.

Example: the fuel economy of a car. In the USA it's usually measured in miles per gallon. It could also be given as gallons per mile, as in many other countries it is given as litres per 100 km. A gallon is a unit of volume, so it has units of length cubed. (You compute volume by multiplying three length measurements, so the result has units of length cubed).

The mile, of course, is just a unit of length. So if you divide gallons by miles you get units of length squared. Does this have a physical meaning? Actually, it does! It's the cross sectional area of the trough of gasoline the car would have to scoop up to continue moving.

Edited to add: As an example, a car that gets 30 mpg would have to scoop up a trough of gasoline with a cross-sectional area of 0.0784 mm^2. That's a square 0.28 mm on a side. Doesn't seem like much, but it adds up.

The same works for electric vehicles, which are rated in units of miles per kilowatt-hour, or kilowatt-hours per 100 km. A kilowatt-hour is a unit of energy, which has basic units of kg m²/s², also known as the joule. (1 kWh = 3.6 million joules.) If you divide that by units of distance, you get kg m/s², which happens to be the newton, the unit of force. In other words, the mileage rating for an electric car is equivalent to the physical force needed to overcome drag and keep the car going. (This also includes some electrical and mechanical losses that appear as "virtual" drag in the final result.)

There are all sorts of other examples like these; physics can be a lot more intuitive than many people think.

[DataCable]

Except, of course, I don't think I've ever seen one of those models in the flesh, so I've only seen a 2d representation or image of a 3d model of a 4d concept...

How about 2D photographs of 3D immersions of 4D Klein Bottles?

[smartcooky]

Except, of course, I don't think I've ever seen one of those models in the flesh, so I've only seen a 2d representation or image of a 3d model of a 4d concept...

How about 2D photographs of 3D immersions of 4D Klein Bottles?

Well, I've seen 3d models of a 4d hypercube.  Sorta like the usual 2d representation of a 3d wireframe cube.  Except, of course, I don't think I've ever seen one of those models in the flesh, so I've only seen a 2d representation or image of a 3d model of a 4d concept...

How about an animated 2d image of a 3d model of a 4d concept being rotated in 2 dimensions of a 4D space...?

[gillianren]

Anyhow, I sorta figured that's what they meant.  Although that sentence bugged me.  (Not as much as a later one, in the second book...about Charles Wallace just "needing to adapt...!")

If I'm right and it's Swiftly Tilting Planet, a reference in the second book is earlier.  It's A Wrinkle in Time, A Wind in the Door, and then A Swiftly Tilting Planet.  And then Many Waters and so forth.  I don't actually like A Swiftly Tilting Planet, though, largely because I don't like Charles Wallace all that much.

As to the animation--mind?  Blown.

[Glom]

the sticky wicket for most non-mathematicians and non-physicists is the notion that an abstract arithmetic operation has meaning to a real-world object such as a unit of time.

The really interesting thing is how often you can find a real-world meaning to the units on some set of measurements.

Example: the fuel economy of a car. In the USA it's usually measured in miles per gallon. It could also be given as gallons per mile, as in many other countries it is given as litres per 100 km. A gallon is a unit of volume, so it has units of length cubed. (You compute volume by multiplying three length measurements, so the result has units of length cubed).

The mile, of course, is just a unit of length. So if you divide gallons by miles you get units of length squared. Does this have a physical meaning? Actually, it does! It's the cross sectional area of the trough of gasoline the car would have to scoop up to continue moving.

Edited to add: As an example, a car that gets 30 mpg would have to scoop up a trough of gasoline with a cross-sectional area of 0.0784 mm^2. That's a square 0.28 mm on a side. Doesn't seem like much, but it adds up.

The same works for electric vehicles, which are rated in units of miles per kilowatt-hour, or kilowatt-hours per 100 km. A kilowatt-hour is a unit of energy, which has basic units of kg m²/s², also known as the joule. (1 kWh = 3.6 million joules.) If you divide that by units of distance, you get kg m/s², which happens to be the newton, the unit of force. In other words, the mileage rating for an electric car is equivalent to the physical force needed to overcome drag and keep the car going. (This also includes some electrical and mechanical losses that appear as "virtual" drag in the final result.)

There are all sorts of other examples like these; physics can be a lot more intuitive than many people think.

I was having trouble understanding what you meant by trough but I think I have it.

Imagine a car is like a train running on a third rail. Except instead of a rail of steel carrying a current, it is a gully filled with petrol that miraculously doesn't evaporate. Instead of a contact shoe, it has a suction inlet that dips into the petrol and is shaped to spade the gully and as the car moves, the inlet sweeps the gully leaving it dry behind and sucks in the petrol it sweeps. The thirstier the car, the bigger this gully needs to be to satisfy the requirements. The cross sectional area of this gully is mathematically equivalent to the consumption.

[JayUtah]

It's the cross sectional area of the trough of gasoline the car would have to scoop up to continue moving.

Neat!  And true!

[JayUtah]

As to the animation--mind?  Blown.

The mind-blowy thing is the realization that the spaces between a face on the inner cube and its corresponding face on the outer cube are degenerate cubes.  You can "project" a 4D cube into three dimensions using the same technique as a 3D cube into two dimensions, knowing that some of the square faces will be distorted by the projection (i.e., perspective).

Consider a train track where the ties are spaced along the track the same distance as the width between the rails.  Thus looking straight down, the ties appear to form a line of adjacent squares.  But looking ahead from the locomotive cab, the squares appear to be trapezoids, the rails converging to their vanishing point and the farther ties appearing shorter and more closely spaced.  Now imagine rolling forward and keeping your eye on one set of ties.  It first appears as a trapezoid, but then gradually becomes more square-like as you approach.  Looking through the glass bottom of your magic locomotive, there will be one instant where you see it as the square before it recedes into the distance to become a trapezoid again.

So watch the animation anew and see where the inner cube is briefly a perfect cube before it's distorted into a cube-trapezoid.

[Sus_pilot]

the sticky wicket for most non-mathematicians and non-physicists is the notion that an abstract arithmetic operation has meaning to a real-world object such as a unit of time.

The really interesting thing is how often you can find a real-world meaning to the units on some set of measurements.

Example: the fuel economy of a car. In the USA it's usually measured in miles per gallon. It could also be given as gallons per mile, as in many other countries it is given as litres per 100 km. A gallon is a unit of volume, so it has units of length cubed. (You compute volume by multiplying three length measurements, so the result has units of length cubed).

The mile, of course, is just a unit of length. So if you divide gallons by miles you get units of length squared. Does this have a physical meaning? Actually, it does! It's the cross sectional area of the trough of gasoline the car would have to scoop up to continue moving.

Edited to add: As an example, a car that gets 30 mpg would have to scoop up a trough of gasoline with a cross-sectional area of 0.0784 mm^2. That's a square 0.28 mm on a side. Doesn't seem like much, but it adds up.

The same works for electric vehicles, which are rated in units of miles per kilowatt-hour, or kilowatt-hours per 100 km. A kilowatt-hour is a unit of energy, which has basic units of kg m²/s², also known as the joule. (1 kWh = 3.6 million joules.) If you divide that by units of distance, you get kg m/s², which happens to be the newton, the unit of force. In other words, the mileage rating for an electric car is equivalent to the physical force needed to overcome drag and keep the car going. (This also includes some electrical and mechanical losses that appear as "virtual" drag in the final result.)

There are all sorts of other examples like these; physics can be a lot more intuitive than many people think.

I was having trouble understanding what you meant by trough but I think I have it.

Imagine a car is like a train running on a third rail. Except instead of a rail of steel carrying a current, it is a gully filled with petrol that miraculously doesn't evaporate. Instead of a contact shoe, it has a suction inlet that dips into the petrol and is shaped to spade the gully and as the car moves, the inlet sweeps the gully leaving it dry behind and sucks in the petrol it sweeps. The thirstier the car, the bigger this gully needs to be to satisfy the requirements. The cross sectional area of this gully is mathematically equivalent to the consumption.

[Thread hijack]That's one of the ways the New York Central Railroad and the Pennsylvania Railroad replenished the water in steam locomotive tenders (I'm not just a pilot, as Jay can tell you).  [/Thread hijack]

[ka9q]

The cross sectional area of this gully is mathematically equivalent to the consumption.

Exactly! And the small size of this cross section for a typical car demonstrates just how energetic gasoline really is, and why it's so difficult to find something to replace it.

The computer in my Nissan Leaf typically shows around 3.9 miles/kWh. (I'm not sure if this is DC energy from the battery or AC wall socket energy. Probably DC.) This works out to 1.7 mm/joule, or an equivalent drag force of 574 newtons (about 129 pounds force).