Over the past few years, there has been increasing excitement about travelling to Mars, and the
possibility of establishing a human colony on the red planet.
With recent breakthroughs in rocket engineering and the onset of a modern space race, it seems
almost inevitable that there will be humans living there within the next decade.
This got me thinking about what a future civilization might look on Mars once humans become an established
species there. What will our homes look like?
How will we travel from one place to another? How will we grow crops?
And how long will it take for Disney to open the first Martian Disneyland.
Even though that one’s a bit of a joke, I started to wonder what a theme park on Mars
might be like, or if it would even be fun to visit a theme park in low gravity.
Many amusement rides utilize gravity in one way or another, but the gravitational acceleration
on Mars is 62% lower than it is on Earth because of the planet’s smaller size.
A falling object on Earth will accelerate at a rate of 9.81 m/s2, but a falling object
on Mars will only accelerate at a rate of 3.71 m/s2.
I decided that it might be interesting to explore how this would impact the physics
of theme park rides by taking a look at one of the most common attractions: The roller coaster.
Roller coasters are one of the best examples
to use because they are driven entirely by gravity and the law of conservation of energy.
Any roller coaster always starts off by giving the train a certain amount of energy in one
of two ways. Either with a lift hill, where the train gains
gravitational energy according to the equation Eg = m*g*h, where m is the mass of the train,
g is acceleration due to gravity, and h is the height above a reference point.
Or with a launch system, where the train gains kinetic energy according to the equation Ek
= ½*m*v2, where m is the mass of the train and v is the speed of the train.
The energy is then converted back and forth between gravitational and kinetic energy as
the train travels around the course, going up and down through the various elements.
Energy is gradually lost due to friction and drag as the train travels along the track,
and the train slowly loses speed throughout the ride.
The speed may also be reduced with braking systems to dissipate energy more quickly,
as is the case for mid-course brake runs, trim brakes, and the final brake run at the
end of a ride. If we consider a generic roller coaster with
a lift hill and a first drop of height h, then we can equate the gravitational energy
at the top of the ride to the kinetic energy that it will have at the bottom in order to
calculate the speed of the train. Since the mass of the train, m, is present
in both equations, it will cancel out on both sides, indicating that the speed of the ride
is not affected by the weight of the train or the passengers.
This only happenes because we are neglecting energy losses here, otherwise there will be
an additional term to deal with and we cannot cancel out mass.
Since we are only looking at one part of the ride by itself, we are ok to neglect energy
losses without introducing too much error so that we can keep the calculation simple.
If we rearrange our equation to solve for v, then we arrive at a simple expression for
the speed of the train as a function of the drop height and gravitational acceleration.
This roller coaster has a drop that is 47 m tall, and since we are currently on Earth,
we will plug in g as 9.81 m/s2. This gives a final speed at the bottom of
the drop of 30.4 m/s, or about 109 km/h. If we want to know the acceleration of the
train during the drop, then we can draw a vector for the acceleration due to gravity,
and we can break it up into two components: 1 parallel to the track and 1 perpendicular
to the track. The vector component parallel to the track
represents the acceleration of the train, and it is equal to g*sin(θ), where θ is
the angle of the track. The steeper the track is, the faster the acceleration
will be, and if θ is equal to 90 degrees, then the train will essentially be in a freefall.
For this coaster though, the maximum track angle is 65 degrees, and we can calculate
the acceleration as 8.9 m/s2, or about 0.9 g’s.
At this rate, the train will take about 3.5 seconds to reach its top speed at the bottom
of the drop, which would look something like this from the view of the passengers.
The maximum g-force that the passengers will experience at the bottom of the drop is 4.2
g’s, which is calculated using the principle of centripetal acceleration.
Centripetal acceleration, ac, is equal to v2/r, where v is the speed of the train and
r is the radius of curvature of the track. If we take acceleration due to gravity and
add on v2/r at that same point, then we will have the total g-force that the passengers
feel as the train pulls out of the drop. In this case, the speed of the train is 30.4
m/s and the radius of curvature is 29 m, resulting in a total acceleration of 41.6 m/s2 or 4.2
g’s. Similar calculations can be applied to the
vertical loop, and we will find that the maximum g-force at the bottom of the loop is 3.5 g’s,
and the minimum g-force at the top of the loop is 0 g’s.
This means that the passengers will experience a brief moment of weightlessness as the train
passes around the apex. I’m not going to dive any deeper into the
physics here, but if you want to know more about how vertical loops work and how they
are designed, then I recommend checking out my video on non-circular loops which explains
the underlying physics and calculus. But now that we have a good basic understanding
of how roller coasters work, let’s see what would happen if we built this exact same roller
coaster on the surface of Mars. Although the train would be lifted to the
same height of 47 m, it will gain 62% less gravitational energy than it did on Earth
because the gravitational acceleration is 62% lower here.
This means that the train will be travelling slower at the bottom of the drop after the
gravitational energy is converted to kinetic energy, however the decrease in speed is not
linearly proportional to the decrease in energy. As we saw before, the speed at the bottom
of the drop can be calculated using v = sqrt(2*g*h), where speed is proportional to the square
root of g. Plugging in our height of 47 m and gravitational
acceleration of 3.71 m/s2, we find that the maximum speed is 18.7 m/s, or 67 km/h.
That’s about 39% slower than the maximum speed on Earth, which comes from the fact
that 1 minus the square root of g on Mars divided by g on Earth is equal to 0.39.
This also tells us that the train will take about 61% longer to reach the bottom of the
drop, at just under 6 seconds, which would look something like this.
The simulation of the roller coaster on Mars probably looks unnatural to you, and it might
even make you feel a little uncomfortable since we are accustomed to the gravity on
Earth. But not only does the coaster look unnatural,
it would also feel unnatural to ride because of the lower forces.
The g-force values themselves would actually be the same as on Earth, with 4.2 g’s at
the bottom of the drop, 3.5 g’s at the bottom of the loop, and 0 g’s at the top of the
loop, however g-force is relative to gravitational acceleration.
The base measurement of 1 g on Earth corresponds to an acceleration of 9.81 m/s2, but 1 g on
Mars corresponds to an acceleration of 3.71 m/s2.
And because force is equal to mass times acceleration, the passengers will experience 62% less force
throughout the entire ride compared to the identical roller coaster back on Earth.
It’s interesting to note, though, that if you feel weightless on a certain part of the
ride on Earth, then you will also feel weightless at the same point on Mars because 0 g’s
still means zero acceleration and zero force. But since roller coasters on Mars would produce
lower forces overall and they would also travel quite a bit slower than their Earth counterparts,
they probably would not be very exciting for humans raised on Earth, aside from being located
on another planet. This would not be the case for humans raised
on Mars though, since they would be accustomed to living in lower gravity.
If a person born and raised on Mars were to ride a roller coaster on their home planet,
they would perceive the exact same g-forces as a person born and raised on Earth riding
the same roller coaster on their home planet. The difference in gravity only becomes apparent
when we have humans travelling from one planet to another, since the actual force magnitudes
would be quite different. And while a human from Earth might find a
Martian roller coaster to be somewhat boring, it would be extremely dangerous for a human
from Mars to visit a theme park on Earth because the forces would be more than 2.5 times what
they are used to, and they would probably black out on just about any roller coaster.
This is something to keep in mind for the future coaster enthusiasts on Mars who want
those interplanetary credits. Now, at this point, you may be wondering about
those launched roller coasters that I briefly mentioned earlier, and what would happen if
we built one of those on Mars. Most of the launch systems that are used for
roller coasters rely on the principles of pneumatics, hydraulics, and electromagnetism
instead of gravity, and these systems would be able to launch a train to the same speed
on Mars as they do on Earth. Since the amount of kinetic energy that they
deliver is not dependent on gravitational acceleration, using a launch system on Mars
is one way that we could create a faster ride and a more intense experience.
However, we will probably never see a powerful strata-launched coaster like Red Force or
Top Thrill Dragster on Mars, since the ride would need to be more than 2.5 times taller
in order to convert all the kinetic energy into gravitational energy.
If we were to build an exact replica of Top Thrill Dragster, the launch would accelerate
the train nearly 4 times faster than Mars gravity, and it would have enough speed to
reach a height of 340 m. It simply would not be practical to build
a roller coaster this tall on any planet, and I think it’s far more likely that we
would see a more traditional roller coaster with an added launch like the Incredible Hulk
at Universal Orlando. In this case, the launch speed could even
be adjusted to provide varying levels on intensity depending on whether the passengers are accustomed
to Earth gravity or Mars gravity. So now that we’ve taken a look at how gravity
would affect the physics of roller coaster on Mars, there’s only one more thing that
we need to consider before wrapping up the video, and that’s the atmosphere.
At the surface of Mars, the atmosphere is about 98% less dense than it is on Earth,
and it is not suitable for humans to breathe. The environment is also very dry with frequent
dust storms, which would make it difficult to operate and maintain a roller coaster outdoors.
The wind forces are actually quite negligible on Mars because the air is so thin, even though
the average wind speed is higher than it is on Earth, but the extremely fine dust would
cover the entire ride and possibly cause moving parts to seize.
For these reasons, it would likely be necessary for any Martian roller coaster to be built
inside a pressurized building or tunnel with similar atmospheric conditions to Earth.
Unfortunately, this would detract from the novelty of riding a roller coaster on another
planet, but it would be necessary to protect the passengers and the ride itself from the
harsh environment. And of course, this would have zero impact
on the low gravity physics of the ride, so I’m confident that the experience would
still be something out of this world.
Hey everyone, I hope you enjoyed today’s more abstract topic about roller coasters on Mars.
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