Falling Friday
Thread Starter
It's QQ thankyouverymuch
iTrader: (39)
Joined: May 2007
Posts: 19,721
From: San Jose
Car Info: 2011 SWP WRX Hatch
Call me Pebbles
iTrader: (11)
Joined: Nov 2011
Posts: 5,908
From: I do all my own physics.
Car Info: Moist
There are a number of things wrong with that scene. Give me a little bit..
Avs fought which is all I can really hope for. Saturday starts their playoffs.
Work.. tomorrow.. que?
Lazy public employee.
Avs fought which is all I can really hope for. Saturday starts their playoffs.
Lazy public employee.
Call me Pebbles
iTrader: (11)
Joined: Nov 2011
Posts: 5,908
From: I do all my own physics.
Car Info: Moist
Total Recall’s “fall” or gravity elevator. God, where to start..
We’ll start with the idealized model and work our way into the weeds.
The ideal model is as I discussed yesterday: the Earth as a perfect sphere of uniform density. From that, we derived that the gravitational force would decay linearly as with decreasing distance from the center of the Earth. This model would have the elevator dynamics be akin to a weighted spring. At the top of the tunnel, the spring is compressed. When released, the spring extends imparting a net positive force on the weight resulting in a positive acceleration (speeding up). At some distance, the spring reaches an equilibrium such that it is no longer exerting a force on the weight as acceleration is 0. The spring extends beyond its equilibrium and imparts a net negative force on the weight resulting in negative acceleration (slowing down). To first principles, this model is fine as we can simply replace the spring with gravity as the source of the imparting force. This is a well-known model of simple harmonic motion that has been widely examined. Applying this to the gravity elevator, the sensation would be the same as any other elevator descending only more exaggerated. As the car begins to move, the force of the acceleration of the car would approach and eventually be equal to the gravitational force so the result be would a gradual approach to weightlessness. About a third of the way into the trip, the force of gravity would be about half of that at the surface. Halfway through, there would sensation of weightlessness and the direction of the net force changes direction. About two-thirds through, the force of gravity would again be about half, but now it is in the opposite direction. As the trip ends, the sensation would approach that of normal gravity. This is the idealized case.
Where this begins to break down is the modeling the Earth as a uniform density sphere such that the gravitational force below the surface decays linearly with decreasing radius. Reality is the density of the Earth is very much NOT uniform. If this were the case, the plot of density as a function of the radius would be a flat horizontal line. In fact, the plot shows that there are very distinct regions of density that actual increase in density as radius decreases!
Piece-wise functions are messy and time-consuming to model. To rough approximations, it resembles a polynomial for density as a function of radius like:
(A) ρ(r)=A-r-r^2
Using that approximation for the model of density as a function of radius and picking up at (5) from yesterday, the mass of the Earth as a function of radius also becomes a polynomial of the form:
(B) M(r)=ar^3-br^4-cr^5
Moving ahead and plugging that into (3) from yesterday, (6) for the gravitational force as a function of radius is no longer linear, but is a third-order polynomial:
(C) g(r)=αr-βr^2-γr^3
The graph of g(r) versus r would look something like this:
From that, we see that the downward gravitation force would gradually decrease as they approach the center and gradually increase in the opposite direction upward as they approached the surface on the other end--either side of the center, the gravitational force would be directed inward to the center. Some flaws in the orientations depicted in the movie: during the first half of the descent, there would not be a sensation of normal gravity as depicted, and weightlessness would not be momentary nor sudden. It would be a gradual decrease, a fairly protracted period of "low gravity" as approximately 75% of the trip would be at or below 0.5 g and ending with a gradual increase back to normal gravity.
That’s one problem, but the next one is much more catastrophic.
Looking at the trip durations, the movie states it takes about 17 minutes to go from England to Australia. Going through the center of the Earth under the influence of only gravity, traveling the diameter of the Earth would take approximately 42 minutes. This period is assumed to be much shorter with the model of the interior densities increasing with decreasing distance from the center. As the crow flies, England and Australia are 9442 miles apart or rough 136.5 degrees apart assuming a uniform circumference (which it’s not). That means a straight shot through the earth is still a trip of about 5450 miles or about 8770 km. This is about two-thirds of the diameter of the Earth. Going back to uniform density, we see it took 42 minutes to go through whereas two-thirds of that is still 28 minutes which is still a fair amount more than in the movie. Clearly depicted in the movie, the path curves around the core in which they lose the benefit of the core’s increased density contributing to the acceleration to make up the difference. As such, to get to the 17-minute mark, the car would need to be propelled. If it is propelled, it is no longer a gravity elevator, but just... an elevator. The propelled elevator would need to accelerate a rate much greater than under the influence of only gravity. In that event, Colin Farrell and his book would then experience "gravity" in the upward direction.
This leads me back to my first noted flaw. Not only did they get the scene wrong, but they got it backward.
We’ll start with the idealized model and work our way into the weeds.
The ideal model is as I discussed yesterday: the Earth as a perfect sphere of uniform density. From that, we derived that the gravitational force would decay linearly as with decreasing distance from the center of the Earth. This model would have the elevator dynamics be akin to a weighted spring. At the top of the tunnel, the spring is compressed. When released, the spring extends imparting a net positive force on the weight resulting in a positive acceleration (speeding up). At some distance, the spring reaches an equilibrium such that it is no longer exerting a force on the weight as acceleration is 0. The spring extends beyond its equilibrium and imparts a net negative force on the weight resulting in negative acceleration (slowing down). To first principles, this model is fine as we can simply replace the spring with gravity as the source of the imparting force. This is a well-known model of simple harmonic motion that has been widely examined. Applying this to the gravity elevator, the sensation would be the same as any other elevator descending only more exaggerated. As the car begins to move, the force of the acceleration of the car would approach and eventually be equal to the gravitational force so the result be would a gradual approach to weightlessness. About a third of the way into the trip, the force of gravity would be about half of that at the surface. Halfway through, there would sensation of weightlessness and the direction of the net force changes direction. About two-thirds through, the force of gravity would again be about half, but now it is in the opposite direction. As the trip ends, the sensation would approach that of normal gravity. This is the idealized case.
Where this begins to break down is the modeling the Earth as a uniform density sphere such that the gravitational force below the surface decays linearly with decreasing radius. Reality is the density of the Earth is very much NOT uniform. If this were the case, the plot of density as a function of the radius would be a flat horizontal line. In fact, the plot shows that there are very distinct regions of density that actual increase in density as radius decreases!
Piece-wise functions are messy and time-consuming to model. To rough approximations, it resembles a polynomial for density as a function of radius like:
(A) ρ(r)=A-r-r^2
Using that approximation for the model of density as a function of radius and picking up at (5) from yesterday, the mass of the Earth as a function of radius also becomes a polynomial of the form:
(B) M(r)=ar^3-br^4-cr^5
Moving ahead and plugging that into (3) from yesterday, (6) for the gravitational force as a function of radius is no longer linear, but is a third-order polynomial:
(C) g(r)=αr-βr^2-γr^3
The graph of g(r) versus r would look something like this:
From that, we see that the downward gravitation force would gradually decrease as they approach the center and gradually increase in the opposite direction upward as they approached the surface on the other end--either side of the center, the gravitational force would be directed inward to the center. Some flaws in the orientations depicted in the movie: during the first half of the descent, there would not be a sensation of normal gravity as depicted, and weightlessness would not be momentary nor sudden. It would be a gradual decrease, a fairly protracted period of "low gravity" as approximately 75% of the trip would be at or below 0.5 g and ending with a gradual increase back to normal gravity.
That’s one problem, but the next one is much more catastrophic.
Looking at the trip durations, the movie states it takes about 17 minutes to go from England to Australia. Going through the center of the Earth under the influence of only gravity, traveling the diameter of the Earth would take approximately 42 minutes. This period is assumed to be much shorter with the model of the interior densities increasing with decreasing distance from the center. As the crow flies, England and Australia are 9442 miles apart or rough 136.5 degrees apart assuming a uniform circumference (which it’s not). That means a straight shot through the earth is still a trip of about 5450 miles or about 8770 km. This is about two-thirds of the diameter of the Earth. Going back to uniform density, we see it took 42 minutes to go through whereas two-thirds of that is still 28 minutes which is still a fair amount more than in the movie. Clearly depicted in the movie, the path curves around the core in which they lose the benefit of the core’s increased density contributing to the acceleration to make up the difference. As such, to get to the 17-minute mark, the car would need to be propelled. If it is propelled, it is no longer a gravity elevator, but just... an elevator. The propelled elevator would need to accelerate a rate much greater than under the influence of only gravity. In that event, Colin Farrell and his book would then experience "gravity" in the upward direction.
This leads me back to my first noted flaw. Not only did they get the scene wrong, but they got it backward.
Thread Starter
It's QQ thankyouverymuch
iTrader: (39)
Joined: May 2007
Posts: 19,721
From: San Jose
Car Info: 2011 SWP WRX Hatch
Say it's not propelled and they simply missed reality by 11 minutes.
I'd say that's pretty damn close.
Maybe time moves slower in the future.
I'd say that's pretty damn close.
Maybe time moves slower in the future.
Churro Aficionado
iTrader: (38)
Joined: Feb 2008
Posts: 54,661
From: IG - @thomas.teammoist
Car Info: IG - @TEAMMOISTOFFICIAL
11 minutes is pretty far off... if it was seconds, or a minute.. maybe.
Given a season of Physics with Samir, on a car forum, on a slow rainy Friday... could come up with the correct answer... says the writers didnt do their research.
Given a season of Physics with Samir, on a car forum, on a slow rainy Friday... could come up with the correct answer... says the writers didnt do their research.
Thread Starter
It's QQ thankyouverymuch
iTrader: (39)
Joined: May 2007
Posts: 19,721
From: San Jose
Car Info: 2011 SWP WRX Hatch
Considering that Samir's calculations differ from WolframAlpha's... I think 11 minutes is acceptable.
The Science of Total Recall?Wolfram|Alpha Blog
And I'm proud that Samir didn't just ****paste that.
The Science of Total Recall?Wolfram|Alpha Blog
And I'm proud that Samir didn't just ****paste that.
Call me Pebbles
iTrader: (11)
Joined: Nov 2011
Posts: 5,908
From: I do all my own physics.
Car Info: Moist
Relativistic effects don't really surface until you're within about an order of magnitude of the speed of light or speeds greater than 0.1c (30,000 kilometers per second). But, we’ll pull at that string anyway.
*cracks knuckles*
Assume the best case scenario: Australia to England (5450 miles) in 17 minutes. Speed (assuming one-dimension) being the ratio of distance to time, that puts them at an average speed of 19,235 miles per hour. 19,235 miles per hour is roughly 8600 meters per second or 0.0000287 times the speed of light. Yes, the speed of light is slower through a non-zero density medium, but we did also assume the tunnel to be evacuated. At the blazing speed of roughly 0.003% of the speed of light, time-dilation correction factor, γ (gamma, not y), would be:
γ = 1/√(1-(v^2/c^2)) = 1/√(1-((8,600 m/s)^2/(299,792,458 m/s)^2) ) = 0.9999999992
Taking into that correction into account, and assuming the 17 minutes was measured by Colin Farrell and his book, a stationary clock on either end of the tunnel would have shown that approximately 17.000000014 minutes would have elapsed.
Here’s one for you to chew on: time does not exist and is but an arbitrary assignment of values.