Pretend our future colonists build a large round 2km diameter train track on the great plains of Mars that is sloped (angled) inward to artificially increase the gravity felt by the occupants living onboard the house sized train cars speeding along at the rails a constant velocity.

Q: Using 3.8m/s^2 for Martian gravity what is the train car speed and track angle required to simulate Earth’s 9.81m/s^2 gravity? And what are the formulas to derive this answer?

Mars Gravity Source: “An Introduction to Physical Science” by James Shipman, Jerry D. Wilson, and Aaron W Todd

This is not a homework assignment, I'm just doing it for fun in writing a sci-fi screenplay

A centripetal acceleration (horizontal) of 9.04 m/s^2, combined with the existing (vertical) Mars gravity acceleration of 3.8 m/s^2, will reasult in a resultant acceleration of 9.81 m/s^2.

Require that V^2/R = 9.04 m/s^2

V = 134 m/s = 300 mph

It does not seem practical. You'd need a bullet train. Why bother? Enjoy the low g.

Good sci-fi strives to limit itself to the laws of physic within the boundaries of future science and technology; within the relm of my future world of science, abundant clean energy is reality and so powering a railed community is possible.

"Why bother?" Because our muscles and bone denisity will become dramtically weak over extended time in low-g; our Martian colonists would deteriorate to a point where they cannot return to Earth and at that point they are no longer able to "Enjoy the low g." because it no longer seems "low" to them, it's now become normal.

I will have a low-g track used for senior citizen assisted living; the hospital will be on Mars gravity to make recovery easier and also moving patients around; also seniors in their final years may opt to live 100% on Mars gravity.

Regarding the math, can you help me understand how you arrived 9.04?

I can determine the proper angle for a frictionless bank if I know Velocity:
R = 152.336 meters
Bank Angle (A) = DEGREES(ATAN(V^2/(R*9.81))) = 85.24243368

And I can determine Velocity if I know the bank Angle:
Velocity = SQRT(R*9.81*TAN(RADIANS(A))) = 134

But I cannot figure out how to correctly determine them without knowing what the other value is.

Thanks for your help.

I forgot to mention that can reduce the speed by making the radius smaller, but studies show that at a certain point it becomes uncomfortable for long-term living; it seems that nausia effects mildly increase below -.5 mile diameters and the nausia increases as the diameter decreases from there; so I intend to keep my minimum diameter at 1km and build my track rings outward from there up to 2km diameters. If the city grows larger than this initial set of community rings (perhaps called communirings, commrings or c-rings) then a new ringset would be built with it's center point located slighly more than 2km from the first center point so that it can also grow to a diameter of 2km.

I searched around to find some links regarding the bone loss, but I'm not allowed to link it here. Just google this long sentence here:

Tests indicate bone loss rates of up to 1.5% a month johnson space center

and you'll find it right away.

In just a few years the colonist could not return back to Earth.

I see that 9.04 that you provided comes from SQRT(9.81^2 - 3.8^2).

But I think that your velocity calculation is wrong because velocity should be the SQRT(R * 9.04) and that would be 67.25m/s, 242.09kph, 150.43mph.

Please let me know if you concur.

The angle will always be the same; it's =DEGREES(ACOS(3.8/9.81))

Thanks again for your help.

Mark

I am positive now; the correct answer to this problem is this:

Extra Gravity Needed (X) =SQRT((9.81)^2 - (3.8)^2) = 9.044119637
V = SQRT(X * 500) = 67.2462624854992m/s = 242.0865449kph = 150.4256051mph
Angle =DEGREES(ATAN(V^2/(500*3.8))) = 67.2096798262262 degrees

To simulate Earth's gravity on Mars, we need to determine the train car speed and track angle required. Here is how we can calculate them:

1. Let's start with the formula for centripetal acceleration (ac) for an object moving in a circular path: ac = v^2 / r, where v is the velocity and r is the radius of the circular path.

2. For the train cars speeding along the track, the net force acting towards the center of the circular path should be equal to the force due to gravity on Earth (mg). So, the net force is given by the equation: m * ac = m * g, where m is the mass of the object.

3. On Mars, the acceleration due to gravity (g) is 3.8 m/s^2, and on Earth, it is 9.81 m/s^2. We can set the Martian net force equation equal to the Earth net force equation to find the relationship between the velocities on the two planets:

m * (v_mars^2 / r) = m * (9.81 m/s^2) ... (eq. 1)

4. The diameter of the train track is given as 2 km, which means the radius (r) is half of that, or 1 km (1000 meters).

5. Solving equation 1 for v_mars^2 gives:

v_mars^2 = (9.81 m/s^2) * r ... (eq. 2)

Substituting the value of r, we get:

v_mars^2 = (9.81 m/s^2) * (1000 m)

Simplifying gives:

v_mars^2 = 9810 m^2/s^2

Therefore, the velocity on Mars would be v_mars = sqrt(9810 m^2/s^2) ≈ 99.05 m/s.

6. Now, let's calculate the angle of the track (θ) required to simulate Earth's gravity.

The equation for the tan of the track angle can be derived as follows:

tan(θ) = v_mars^2 / (r * g_mars)

Substituting the values, we have:

tan(θ) = (99.05 m/s)^2 / ((1000 m) * (3.8 m/s^2))

Simplifying gives:

tan(θ) ≈ 26.04

Therefore, the track angle θ can be found by taking the inverse tangent (arctan) of 26.04:

θ ≈ arctan(26.04)

Using a calculator, we find that θ ≈ 88.5 degrees.

So, to simulate Earth's gravity on Mars using a 2 km diameter train track, the train cars would need to travel at approximately 99.05 m/s, and the track would need to be angled inward at approximately 88.5 degrees.