Fun Fair Physics

The new Equestria Girls special – Rollercoaster of Friendship – comes with a nice physics lesson courtesy of Twilight Sparkle and Sunset Shimmer. Calculating the trajectories of projectiles is a fun exercise, which in its simplest form just needs high school maths, but if you want to go into more detail, it very quickly gets rather complicated. This takes me back to my school days when I killed time during boring classes by programming artillery games on a graphic calculator. These days it’s the sort of question I might ask students in university admissions interviews.

“Do you know what's not rigged? The laws of physics. Assuming no air resistance and a vertical displacement of zero, horizontal displacement equals initial projectile velocity squared times the sine of twice the launch angle divided by the acceleration due to gravity.”

She is correct. Math enthusiasts see below for a derivation.

But I don’t think I’d give her full marks just yet. She comes across as a student who is good at memorising equations, but lacking a good intuition about how to apply this knowledge. A follow on question might be: to fix that to the precision of a bottle neck width, to what level would you need to control the angle and launch speed? Could you do that by hand?

I couldn’t. But perhaps such a skill is among Twilight and Sunset’s superhuman powers. They do seem able to propel a hoop onto a bottle neck multiple times… Only to watch it rebound in a decidedly frustrating fashion.

However, to a girl with a physics notebook in hand and head full of interesting science facts picked up from SnapGab bots, this is nothing that can’t be tackled with another equation. Deriving a result to include the effect of aerodynamic drag is quite impressive. A shame we don’t get to see her working. Does she include both laminar and turbulent flow? Those bot posts must be pretty detailed if they include the drag coefficients for the game rings.

Sadly this improved trajectory calculation does not lead to a better result. At this point you might be inclined to question the initial assumption that the laws of physics are not rigged. But before we invoke Equestrian magic, or expose ourselves to charges of slander and libel from Flim and Flam by inventing tales of foot pedal-controlled electromagnetic repulsion systems, can we explain this with classical mechanics?

An additional variable which Twilight alludes to is the angular momentum. In such a game, it is sensible to set your ring spinning in the horizontal plane, so conservation of angular momentum will ensure it keeps that orientation and lands neatly around the bottle neck instead of bouncing off the side. If you are seized by a manic determination to set the trajectory as precisely as possible, you would set it spinning as fast as your magical powers permit.

But then when it lands on the bottle neck, all that angular momentum has to be dissipated, so it rattles around the neck for a while, and if it has enough energy, it may bounce back off the bottle altogether. It might be more successful strategy to just toss it without so much spin.

Comments

[FanOfMostEverything]

...or expose ourselves to charges of slander and libel from Flim and Flam by inventing tales of foot pedal-controlled electromagnetic repulsion systems...

In my defense, I never said anything about the activation system being pedal-operated. And we do see a ring with much less spin pop off later. Still, excellent point about angular momentum.

[the dobermans]

I wonder how much the nutation of the disks as they come to rest around the bottle necks (and whether they bounce free) is affected by random starting conditions. If the influence is strong, Twilight might be fighting a losing battle.

That is to say, Flim and Flam may have a side deal with Discord.

[Pineta]

4905140
Now imagining an extended courtroom drama... After presenting video evidence, Twilight makes a strong case that the game must be rigged, ruling out every possible alternative explanation with a full analysis.... and goes on to lose as libel laws are not the same as scientific laws.

4905156
If Discord were involved, things would be even less predictable... And both Flim and Flam, and Twilight, would be fighting a losing battle.

[Kris Overstreet]

4905185 It's a carnival game. Of COURSE it's rigged. It doesn't even take much to do it, really; just choose bottles with short necks and place them so that the ring just barely fits between the necks on all sides of any bottle. You can drop the ring on the bottle of your choice to demonstrate that a ringer is possible, but your marks will throw and throw and throw and watch the rings bounce away.

(Of course, Flim and Flam wouldn't leave it there, but just choosing the right bottles is more than enough. if a winning ringer has to rest on at least two other bottles besides the target, the game is rigged.)

[Alto Legato]

x=\frac{v^2 \sin (2 \theta)}{g}

\sin ^{-1}\left(\frac{gx}{v^2}\right)=2 \theta

Assume:

0\leq \theta \leq \frac{\pi}{2}

Thus:

0\leq \sin ^{-1}\left(\frac{gx}{v^2}\right)\leq \pi

Which will give two solutions that sum up to 90 degrees, the smaller angle increases horizontal speed while the bigger one reduces the horizontal speed.

Strategy:
Throw it with a big angle, horizontally and induce some angular momentum.
Or
Throw it with a tilted angle with angular momentum (spin it fast) so that the part of the ring that tilted down can hook up the bottleneck.