28 Feb 2018

In Physics class, when considering power, the minds of the clever brony might be graced by the question, “How much power did Rainbow Dash produce when performing her famous Sonic Rainboom?”

Let’s do the math.

Power done by a force can be defined as the work done by that force divided by the time the force acted.

$P_a = \frac{W_a}{\Delta t}$

In the case of Rainbow Dash’s sonic rainboom, from season 1, episode 16 of *My Little Pony: Friendship is Magic*, we are specifically interested in the power exerted by Rainbow Dash herself. However, that is not the net work done from her flight; when flying in a nosedive, she is also under the influence of gravity and air resistance. Magic horses don’t exist in a vacuum, you know! Thus, where $W_a$ is Rainbow Dash’s work, $W_g$ is the work from gravity, and $W_r$ is the work from air resistance:

$W_{net} = W_a + W_g + W_r$

By the work-energy theorem, we know that this net work is equal to the change in Rainbow Dash’s kinetic energy:

$W_{net} = \Delta \mathrm{KE} = \frac{1}{2} m (v_f^2 - v_i^2)$

However, it is prudent to consider the sapphiric horse’s exact flight pattern. In particular, after ascending to a massive height, coming to a brief halt, she flies directly down. That is, we can begin measuring her path from the beginning of her descent. This has two major consequences for the velocities. First, $v_f$ is equal to the speed of sound, mach-1. We’ll come back to this fact later. Second, her initial velocity $v_i$ is zero as she begins from rest. Rewriting, we see:

$W_a + W_g + W_r = \frac{1}{2} m v_f^2$

Next, we consider the work done by gravity, $W_g$. A usual high school textbook tells you that this work is related to Rainbow Dash’s mass, the height from which she flies, and the Equestrian gravitational constant $g_E$:

$W_g = m g_E h$

Further, basic kinematics tells us that this height $h$ is also related to time as $\frac{1}{2} g_E (\Delta t)^2$ Again rewriting, we find:

$W_g = \frac{1}{2} m g_E^2 (\Delta t)^2$

Finally, to find the net work, we need to compute air resistance. As Francis Sparkle from *Friendship is Witchcraft* episode “Foaly Matripony” would tell you, computing air resistance of a magical horse is gnarly – specifically, “the bad kind of gnarly, when things are gnarled”!

That said, even without computing such a gnarly number, we know that the sign of the work done by air resistance must be negative. (Proving this is trivial and left as an exercise for the reader.) Thus, the presence of air resistance can only increase the work done by our poor little Dashie – which means, critically, that setting it to zero and proceeding will yield the theoretical lower-bound on her work done, and therefore the lower-bound on her power output.

So, we can now solve for the lower bound on Rainbow Dash’s work $W_a$:

$W_a = \frac{1}{2} m v_f^2 - \frac{1}{2} m g_E^2 (\Delta t)^2$

We can substitute this back in to the definition of power:

$P_a = \frac{1}{\Delta t} (\frac{1}{2} m v_f^2 - \frac{1}{2} m g_E^2 (\Delta t)^2)$

Now, it is a matter of estimating or computing values for $\Delta t$, $m$, $v_f$, and $g_E$.

$\Delta t$ is the easiest to compute, under the assumption that the time shown on camera to the viewer is identical to the real-Equestrian-world time. Pausing the video when Rainbow Dash turns direction at the top and when the Sonic Rainboom first goes boom, we see that the rainboom takes about twenty seconds (plus or minus one second) to be performed. Thus, $\Delta t = 20 \; \text{s}$.

$m$ is up next. This is perhaps one of the most difficult terms to estimate, for the simple reason that there is no clear unit of weight shown within canon. Are ponies teensy, like their namesake show suggests, with masses around 30 kilograms as an adult? Are they like human-world horses, clocking in closer to 300 kilograms? Who knows? We won’t be addressing this question in this paper; instead, defer to this analysis by another Equestrian scientist, who estimates a pegasus (as the lightest race) might weigh around 150 pounds, or about 70 kilograms. Given that there is no better reference available, we may as well set $m$ to $70$. The real number, unfortunately, will vary wildly.

$v_f$ is trivial, under the crucial assumption that an Equestrian rainboom truly does occur when breaking the speed of sound, mach-1, the dominant theory among contemporary Equestrian physicists. Thus, consulting a standard physics text – okay, okay, I used Wikifoalia – yields that $v_f$ is around $343$ meters per second.

$g_E$ is puzzling. One option might be to project human notions of gravity onto the planet of Equus, setting $g_E$ to equal $9.8$ meters per second per second. However, it is prudent to remember the derivation of this constant (from Neighton’s Law of Universal Gravitation) depends, crucially, on the mass of the planet and the elevation. Elevation should not be a huge issue here; while Dashie does fly pie in the sky (as her friend Pinkamena might put it), the difference should be negligible. Further, although the rainboom in question does occur in the sky (in the floating city of Cloudsdale), again there should be negligible difference. The real issue is the mass of the planet – there is no reason to believe that Equus has the same mass as Earth. Indeed, prior to the revelations of *My Little Pony: the Movie*, there was little reason to believe there *was* much of anything outside of the small sovereign nation of Equestria. Still, even with the new knowledge of the outside world and refined cartography, it appears likely that the Equus is somewhat less massive than Earth; therefore, like when ignoring air resistance, we can set $g_E$ to Earth’s gravitational constant to reveal a more or less decent lower-bound on Rainbow Dash’s power. Indeed, given the new revelations, perhaps coupled with some long-time fanon and integration from the Equestria Girls parallel universe, it is possible that Equus could be nearly identical to Earth in size. In any case, we let $g_E$ equal $9.8 \; \frac{\text{m}}{\text{s}^2}$

From here, we have enough information to compute a lower bound on Rainbow Dash’s power output. Substituting into our equation for Rainbow’s power, we find:

$P_a = \frac{1}{\Delta t} (\frac{1}{2} m v_f^2 - \frac{1}{2} m g_E^2 (\Delta t)^2$

$= 138657.75 \; \text{W}$

$= 140 \; \text{kW}$

However, like air resistance computation, the kilowatt is a gnarly unit when considering Rainbow Dash’s power. Instead, let’s use the conversion factor of 735.5 watts, to finally yield that Rainbow Dash, one horse, produces…

189 horsepower!

…

Special shout out to “My Little Pony Physics Presentation” which examines this scene from another light.