Trip the Light Fantastic

This is another item from my list of ideas, based on a news article from a couple years ago: Idiocy On Ice: Speed Skaters Believe They Go Faster Wearing Blue. It should be pretty obvious that the color of the speed skating suits will have no effect on their speed, beyond a psychological boost, but I wondered what kind of advantage could be made from wearing blue over other colors. The first thing I thought of was another news story from the last few years, the LightSail spacecraft.

Light is made up of photons, which carry momentum. When light hits an object, that momentum can give the object a push. Rather than carry its own fuel, the LightSail is pushed by sunlight to accelerate. The same principle could be used with our speed skater and, say, a laser pointer. Blue light is among the most energetic of visible wavelengths, so using a suit that reflects that color and a corresponding laser to aim at it, we can get the best results.

The momentum of a single photon is given by

where h is Planck's constant and λ is the wavelength. Blue light has a wavelength around 420 nm, which gives a momentum of 1.58 x 10^-27 kg m/s. Since we're bouncing the photons off the skater, we actually get twice the momentum compared to absorbing them. To figure out how many photons we need, we have to find the minimum velocity boost that will give an advantage.

In 2018, the men's speed skating medals were decided by 2 milliseconds over a 5000 meter course! The difference in average velocity was 7.24 x 10^-5 m/s. To get the momentum, we need a mass for the skaters, which this page suggests is around 161 pounds. Comparing that to our per-photon momentum shows we need a total of 1.6 x 10^24 photons to make a difference. The energy in each of those photons is simply the momentum multiplied by the speed of light, so the total energy required is 790 kJ. Delivering that over the 6-minute race time gives 2.1 kilowatts. This is what a 2-kilowatt laser looks like:

Any volunteers?

Highweighs

via Wikipedia

This week I thought I'd look into something I've often wondered: Where does the different speed limit for trucks on highways come from?

My assumption has always been that it was a safety issue. Trucks have more mass, and therefore more kinetic energy,

and more momentum,

both of which are factors in collisions. However, each is directly proportional to mass. Typical car masses are around 2,000 kg, and trucks have a maximum total mass of 36,000 kg. That would suggest the speed ratio should be around 1:4.24 for energy, or 1:18 for momentum, while the actual ratio is 1:1.18.

Another thought that occurred to me was stopping distance – Intuitively, I'd expect a truck to take a longer distance to stop, but the more I thought about it, the more I questioned that expectation. The basic model of friction physicists learn is that the force is given by the coefficient of friction between the two surfaces, μ, multiplied by the forces pushing the surfaces together. For a vehicle on a flat road, that's just the force of gravity, mg. We can get the acceleration with Newton's 2nd Law:

According to this, the masses cancel, and the acceleration is μg no matter what you drive.

The answer is that the simple friction model isn't good enough: Tire friction does not increase linearly, but tapers off according to an exponential factor:

We can plug in some numbers to see how well this matches the speed difference. After a bit of algebra, the ratio of stopping distances is

Using ⍺ = 0.7 gives a ratio around 0.6, and ⍺ = 0.9 gives 1.05, putting the stopping distance nearly equal!

The real answer though, is actually neither of those – When finding the image at the top of the post, I noticed the title was "TN environmental speed limit". The actual reason is that trucks (and cars) get better mileage at lower speeds, and since diesel exhaust includes worse pollutants that normal gas, limiting fuel use in trucks is a priority.

Alright Guv'nor!

The other night, we watched the movie Mortal Engines, about a post-apocalyptic society that lives in steampunk-style mobile cities:

YouTube

All the retro-futuristic equipment got me thinking about one of the more popular devices to show, centrifugal governors.

Wikipedia

Some of the steam driving an engine is diverted into this device to make it rotate. The faster it goes, the more the arms swing outward. Depending on the angle, more or less steam is fed into the engine. I was curious to look at some of the physics behind these tools.

We can start by looking at just one of the arms:

The rotational inertia for this design is

which gives the rotational kinetic energy

Meanwhile, the height of the weight is

which gives the gravitational potential energy

We can combine those equations to relate ω and θ, and then use it in the torque equation:

In the case of a steam engine, we want to release pressure if the engine is going too fast, so we could imagine making the torque proportional to the height of the weights by connecting a valve to the arms. Then we get a differential equation for the rotation speed:

where A is the constant of proportionality. If A is negative, then high speeds get slower. As is, this equation would drive toward zero, but it could be engineered to have some minimum speed. I recognize it can get a bit silly at times, with useless gears and cumbersome designs, but I still have a soft spot for steampunk aesthetics, so I enjoyed the film.

Slow, Sarcastic Clap

I mentioned looking forward to coming back to Cardio Drumming at the gym where Marika and I go, but sadly they no longer offer it. Instead, they offer something called Pound Fitness that we've been trying out, which involves using weighted drumsticks without the ball drums. It's got rhythm on my mind, and made me think of another idea I've wondered about for a while: When large groups clap along to music, why does it seem like there are so many who are off-beat?

I suppose the uncharitable answer is that they're bad at keeping time, but I wondered whether the relatively slow speed of sound had anything to do with it. If we imagine the beat as a pulse that leaves the stage at 343 m/s, each audience member will clap when that pulse reaches them. That starts a new beat-pulse that will be out of sync when it reaches other audience members.

The equation for the error is fairly simple:

where R is the distance from the stage to the audience members, c is the speed of sound, and 𝜃 is the angle between the audience members.