Ultimate mathematical question?

Question

I have been pondering this for a couple of years now, and have even asked it to a website or two and a few people. Not once have I ever gotten an answer I felt was properly researched or calculated. I must start by saying that this question is simply to satisfy a bit of a curiosity, no I am not going to try it, so if you need to ask "why does it matter?", just shut up and go the hell away (that will definitely secure a couple "why does it matter" remarks...) Anyhoo, if there were a highway that was perfectly straight and went around the entire earth and connected to itself, and you were driving, say a 4,000 pound vehicle (one could kinda estimate external dimensions from a truck or something for wind resistance and drag), how fast would you have to go before you left the road/shot off into space? Like, slingshotted yourself away from the road due to the curvature of the earth? Is there any way to actually figure this out?

Well, I was going to say that at some point, you would not be able to rely on the speed that your tires on the vehicle would give you and you would need some sort of secondary propulsion, but I ran out of room. I figured your would need a rocket or something affixed to the vehicle in order to continue the acceleration cause reaching the "escape velocity" would probably not happen all of a sudden, and hence, there would be a time where your wheels would leave the ground and you would decelerate and decend. So, I guess to add to the question, figure in a form of secondary propulsion attached to the vehicle, i.e. rocket, jet engine, etc.

Answer 1

This is actually pretty straightforward...let's ask the question this way, how fast would the earth have to spin before its rotation caused objects to hurl upwards at the equator?

Well, you would need the earth to spin rapidly enough that an object at the equator would experience an amount of centripetal force equal to its weight pulling it down, in math terms, this means:

mg=mv^2/R where R is the radius of the Earth, g is accel due to gravity and m (which is irrelevant since it cancels) is the mass of the object

it turns out that v=sqrt[gR], and this means a rotational period (the length of the day) of 1 hr and 24 mins!

Well, the earth isn't rotating quite this fast...but notice where most nations put their launching pads. The US has its launching pad in Florida, about as far south as you can get...and Cape Kennedy is on the east coast of the state, so that rockets are launched in the direction of the earth's rotation?
Why, to make use of the Earth's rotational speed as much as possible. The closer you are to the equator, the faster is the earth's rotational speed (the entire surface rotates in 24 hours, so if you have a longer circumference you have a greater speed), and you want to launch with the earth's motion. good question!!

Answer 2

this question is gold. I definitely have asked myself a matching question earlier, if there is an intrinsic equivalence between axioms (represented by capacity of numbers, entire mathematical description) and phenomena (represented by capacity of words, the philosophical description). it may look glaring that there isn't any connection, numbers are 'purpose', their values are self sufficient from the 'observer' and so as that they are marvelous to describe something impersonally; and words 'subjective', loaded with our very own theory of the international with powerful means to impact our minds. additionally this question finally ends up interior the dualism 'strategies-physique' and an entire 'super distinction' between the mathematical based and philosophical based sciences. Mathematical sciences are in accordance with axioms, prepositions and deductions striving to particular purpose values interior the main financial and good way (which seems to be working thus far). Philosophical sciences count on phenomena, concepts and reports arising a by no capacity ending capacity of assumptions and fashions of information, which unlike arithmetic has a great style of subjectivity. It does not try to particular yet fairly to strengthen our know-how turning it clearer. So arithmetic does not might desire to be more suitable, that is a ideal gadget of verbal replace and representation of know-how. With one in all those powerful gadget we would desire to comprehend the thank you to apply it, that's what philosophy is for, we can sometime comprehend that all of us our know-how is mathematical and our philosophical information, or intelligence, would be clearer than it's going to ever be.

Answer 3

You've certainly looked at the reference, and realize it assumes no friction. Under those conditions, once the truck moving horizontally reaches escape velocity relative to a fixed earth (not relative to the ground), it will be capable of leaving earth's gravity with no further thrust. The mass of the vehicle doesn't affect the velocity, just the amount of energy you need to expend to accelerate it.

No, this is nowhere near the ultimate mathematical question.

Answer 4

Thats a good question.

Needless to say, you'd have to be pretty fast.

It's a fairly simple matter to calculate the orbital speed for an object exactly at the earth's surface,

v[orbit] = sqrt ( r*g[orbit] )

http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html

This turns out to be a whopping

7,911 m/s;

or more than 7 kilometers per second, (if my calculator is correct...!) Of course at this speed air friction would burn you up just like a meteor.

If you want to actually gain some altitude, you need to be going even faster than this. Even neglecting hypersonic air friction, you will still just go into an "eliptical orbit," and eventually come crashing back to where you started, unless you could reach escape velocity.

If you want to take into account air resistance, unfortunately it makes this problem hideously, stunningly complex. Basically, you have to solve simultaneous, nonlinear differential equations:

d²r/dt² + GM / r² - r(dθ/dt)² = 0, and

m r (d²θ/dt²) + CD { r (d²θ/dt)}² exp(-kr) = 0.

If all that's way over your head, don't worry: I have no idea how to solve it either! I think it would pose a real challenge even for a career mathematician. It's out of reach for a humble engineer like me....

~WOMBAT

Answer 5

ESCAPE VELOCITY

v = sqrt(2GM/R)

where:
G is earth's gravitational constant
M is the mass of the earth
R is the radius of the earth.

or 11.2 kilometers per second or 25,950 miles per hour as stated by others.

None of this takes into consideration wind resistance.

Good luck with that.

Answer 6

the question is ill formulated, as to what direction you are driving, wind speed, what is the surface friction or height? How could anyone give a precise answer to this ?

Answer 7

Assuming a circular orbit at a distance equal to the earths radius you can use the equation:
v=sqrt(GM/r)

I got 7906.025 m/s.
Thats pretty fast.
Thats 17 685.2742 mph

Answer 8

This is a very well known number called "escape velocity" Wiki it. It's 11.2 km/s. You have to add your highway speed to earth's rotational surface velocity at the point of liftoff and be faster than this, otherwise you'll end up in an elliptical orbit that will bring you back to earth. Air resistance is neglected.

Now, if you just want to lift off and come back down somewhere, you only need to exceed low earth orbital velocity. That's about 6.9 km/s

Answer 9

About 17,500 MPH travelling west on the equator, about 15,500 MPH traveling east on the equator, about 18,000 +/-MPH going north or south.

For leaving the ground A = g = 9.8m/s^2 = r * omega^2 where omega is the number of radians per second you are travelling (the circumference of the earth is 2 * pi radians). The reason the direction matters is because you are travelling about 1,000 mph east on the equator due to the Earrth's rotation.

Answer 10

Here's a thought that may help in the solution: I would use vectors and compare your question with the same scenario of rotating (swinging) a weight on a string (like the old time sling-shots). Such as how much speed it takes to keep it moving in a circle, to overcome gravity, before it just falls down.
Just a thought ... Hope it helps.