Pages

Saturday, March 28, 2020

Exponential Exposé

With Michigan and most other places in the world sheltering from COVID-19, I thought it might be interesting to talk about exponential growth, and how it relates to disease spread. Disease is transferred from one person to others, so the number of new people who get a disease is related to the number who have it now. We can write that as a differential equation:
N is the number of people with the virus, and the dot indicates the change in time. τ is a time constant that tells how quickly the disease spreads. Solving this equation gives
At t = 0, we have one case, and after every τ, the number increases by a factor of e (= 2.71828).

A couple days ago, the blog Hack-a-Day (which I've mentioned before) had a post about getting realtime data about COVID-19 statistics. I wrote up a Python script to pull the data we're interested in, and make some plots. First, we can look at the total number of US cases over time:
According to the equations above, if we look at the increase from day to day, it should be linearly related to the number of cases on that day. Plotting gives
There's some variation, but overall it's fairly close to the fit line. According to that fit, we have τ = 4, meaning the number of cases increases by e every 4 days. To put that in more relatable terms, that's doubling the cases every 2.8 days!

Looking at the US cases compared to the global total,
we can see that the global rate is beginning to taper off compared to the US, suggesting we should do more to limit the spread!

Friday, March 20, 2020

You Can't Take the Sky from Me

[Yes, I recently rewatched Firefly.]

With social distancing on my mind this week, I started thinking about taking it to an extreme: interstellar travel. Specifically, I wanted to try some calculations with the Tsiolkovsky rocket equation, which applies to any space ship that moves by using conventional rockets.

The idea is this: To get moving, you need to push off of something. On the ground, you can push off the ground or the air, but in space your options are more limited. Rockets accelerate fuel out the back, which pushes the payload forward:
via Wikipedia
The problem is that the more mass you carry, the more fuel you need, but more fuel means more mass. If you work out the particulars, the equation you (or rather, Tsiolkovsky) get is:
This says that the more change in velocity you want, the more fuel you need, but there are diminishing returns.

When my parents were shopping for a camper van, they described their ideal as a "Russian space capsule", so let's take them at their word and imagine attaching rockets to their 2000 Chinook Premier. The manual gives the loaded weight as 10,700 lbs, or 4,853 kg. This is m in the equation above. For the exhaust velocity, ve, we can use the numbers for the (now retired) space shuttle: 4,447 m/s. If our target is the edge of the Solar System, we can relate the travel time to the fuel required:
The dashed line shows the mass of the camper, so to get there in under ~2 months, we need more fuel than cargo.

Of course, there's not much to do at the edge of the Solar System, so how long does it take to get to our nearest neighbor, Proxima Centauri, 4.2 light years away? If we want to get there within a reasonable time, we need to go pretty fast, so we'll use the Special Relativity version of the rocket equation:
Relativity says that as you get closer to light speed, it takes more energy to go any faster. Using the same exhaust velocity from before, the curve looks like this:
For reference, the mass of the Earth is around 6 x 10^24 kg, so burning the entire planet could get our camper there in under 10,000 years!

This is the reason other propulsion techniques have been proposed, like the LightSail I mentioned a couple weeks ago, or Project Orion, which involved accelerating the spacecraft by exploding nuclear bombs behind it. I think I prefer to associate my name with the new shuttle replacement, rather than something that leaves a trail of fallout...

Monday, March 16, 2020

The Error of Slitherin'

[I'm not placing any judgment on snakes' movement technique; I just couldn't resist the Harry Potter pun!]

Along with her obvious chihuahua roots, we suspect Lorna has some terrier in her, due to her dogged (heh) attempts to catch small animals, including (to my mother-in-law's horror) snakes!


During a walk this past week, I saw her lunge forward and I instinctively yanked back on the leash to see this fellow slithering away. I was hypnotized by the rhythmic motion, and was immediately curious how it worked.

Searching a bit online, I found a paper called The mechanics of slithering locomotion, which seemed like exactly what I wanted to know. The idea the paper puts forward is that the friction between the snake and the ground is different depending on which way it slides: There's less grip along the length of the snake than across it. You can see the difference in the pattern of scales:
Figure 1 from the paper
The paper goes into a lot of detail about making the snakes move at different angles and, of course drugging them for pictures, but I was curious if I could make a simple model based solely on the idea of direction-dependent friction.

Suppose the snake moves forward at a constant velocity, while moving sinusoidally in the transverse direction. We can write this as a parametric equation:
where v_x is the forward velocity, and A and ω are the amplitude and frequency of the side-to-side motion. Differentiating twice will give us the acceleration needed to maintain this motion. Of course, since the x-velocity is constant, the acceleration is zero, so turning to y,
That's in terms of the direction lateral to the snake's overall motion, but that's not what we're interested in. We want the acceleration in terms of the snake's body direction. The direction of the snake can be written as
We can plug in the derivative, normalize the vector, then use the equations
to get the components of acceleration parallel and normal to the snake. That gets a bit messy though, so I'll skip to the nifty animation:
As you can see, the forces perpendicular to the body (red) are much larger than the ones parallel (blue), requiring more friction as the paper found.

Saturday, March 7, 2020

Ballistic Bridges

Over the last couple weeks, Marika and I were on a road-trip to Florida for a little relaxation between jobs. Along the way, we crossed several drawbridges, and Marika wondered (theoretically, I hope): How fast do you need to drive to jump an open drawbridge?

To be clear, this is the type of bridge I'm talking about:
via Wikipedia
It turns out, everywhere except the US, this is called a bascule bridge, while the term drawbridge is reserved for the medieval type.

If we assume the car leaves the bridge with a certain angle and velocity, and we ignore air resistance, we can use the equations for ballistic motion:
where g is the acceleration due to gravity, and vx, vy are the x- and y-components of velocity. We don't really care about the time, so we can combine these by setting up some coordinates:
The equation for the path becomes
The minimum velocity then is
Of course, this is just the velocity we leave the ramp with. As the tilt increases, it's harder to climb the bridge, so we need additional velocity:
To get an idea of how fast this actually is, we can take some numbers from one of the most famous bascule bridges, Tower Bridge in London:
At 45°, the total speed required is around 80 mph, which may not be out of reach of our little Ford Focus, but I haven't accounted for air resistance, or the rotation of the car as it moves through the air. Maybe if the University of Florida accepts my postdoc application we can get some experimental data!