Authors of this question: Teresa Paula Coelho Azevedo Perdicoulis, ISR Coimbra & Departamento de Matemática, Universidade de Trás-Os-Montes e Alto Douro, Portugal & Paulo Jorge de Azevedo Lopes dos Santos, INESC-TEC & Departamento de Engenharia Eletrotécnica e de Computadores, Universidade do Porto, Portugal
Santa Claus, in full holiday prep mode, wants to give the reindeer a break and test out the new high-tech Cruise Control for his Reindeer-Mobile. He thus takes the sleigh out, prepare it for a ride, sets the speed to “Jingle all the Way” mode, leans back with his cocoa, and waits for a smooth trip.
But as the sleigh starts moving, something curious happens…
At first, the Reindeer-Mobile takes off quickly, as Dasher does when on a candy cane sugar rush. However, the acceleration of the sleigh soon starts tapering down, and Santa notices that the final velocity of the sleigh reaches only 63.2% of the desired speed he initially set. <<Well, well, that’s a rather odd behavior!!!>> he mutters, after stopping the car to adjust his hat.
As someone who doesn’t give up, Santa decides to activate the high-tech features implemented on the Cruise Control: the Pinecone and the Icicle actions.
- The Pinecone action gives the Reindeer-Mobile a nudge every time its speed is falling behind Santa’s wishes. The further the Reindeer-Mobile is from the target speed, the stronger is Pinecone’s nudge. But, even with Pinecone’s help, the system still seems a little sluggish. So the Icicle action intervenes too.
- As we already know, the icicle grows and melts according to the difference between the desired and the actual speed. Depending on the length of the icicle, the controller then takes action. So, the Icicle action starts adding up all the little differences between the target speed and the actual speed over time, with the intuition that this will make sure the Reindeer-Mobile doesn’t settle for anything less than what Santa asked for. The intuition is that with Icicle’s help, the Reindeer-Mobile would gradually eliminate the tracking error, inching closer and closer to the desired target speed.
Unfortunately, Santa didn’t have time to read the details in the instruction manual. In a hurry, he randomly presses one of the many available buttons — the big red one labeled “Overdamped” in bold, visible letters. Scratching his head, Santa wonders: <<What’s going to happen with this darn high-tech sleigh? Will I still manage to complete my Christmas Eve journey at the desired speed?>> What would you tell Santa?
a. 🎅⚡️🏎️ There are actually no problems: the sledge will instantly reach the exact speed set by Santa, as fast as he says <<Ho, Ho, Ho!>>.
b. 🎅🔧🎯 The Reindeer-Mobile will start off quickly, reaching 63.2% of the target speed—the same percentage it would reach without the Pinecone and Icicle action—within the time it takes for Santa to say his famous catchphrase, <<Ho, Ho, Ho!>> After this initial burst, the Reindeer-Mobile will gradually reduce its acceleration as it gets closer to the target speed. It might take a few more Ho!s, but after about a dozen or so, the Reindeer-Mobile will be effectively travelling at the desired speed.
c. 🚀🎄🌌 As in the previous option, the Reindeer-Mobile will start off quickly, reaching 63.2% of the target speed—the same percentage it would reach without the Pinecone and Icicle action—within the time it takes Santa to say his famous catchphrase, <<Ho, Ho, Ho!>>. However, the Reindeer-Mobile keeps accelerating forever, reaching higher and higher speeds. Uh-oh! This could break the space-time holiday continuum and result in a never-ending Christmas! Maybe not bad, after all…
d. ⏳🏎️📉 The Reindeer-Mobile will exhibit a behaviour where its speed quickly rockets to 90% of the target value set by Santa within the time it usually takes a child to unwrap a present (effectively instantaneously). However, it will take an extremely long time to compensate for the remaining 10% of the target speed.
e. 🎢😂🛷 The Reindeer-Mobile will start bouncing up and down, much like Dasher does on a trampoline, resulting in a guaranteed bumpy ride. Santa will find it hard to drive the Reindeer-Mobile effectively, though the elves might at least enjoy the rollercoaster-like experience!
f. 🍪🥱⛔ Introducing the Pinecone and the Icicle action does not help solving the issue, as they will only cause the sleigh to eventually slow down and stop, like Santa does after having had too many gingerbread cookies!