Beating a dead horse: an analysis of the Bugis Ferrari accident

So I saw this video on Facebook¹: video of accident. Apparently some hardwarezone EDMW dude did some rudimentary analysis that immediately got picked up by the Temasek Times as evidence that the state media is lying to us etc etc.

Let's analyse this for a bit. It's instructive to go through this frame by frame, so for ease of reference I've done it up in a nice convenient package.

Preliminary notions

We assert that:

1. The video maintains a constant frame rate, which is trivial to verify, and

   

2. The timestamp at the bottom is a representative record of realtime. This one is easily determined for correspondence to realtime in playback, but we need to establish that it also corresponds to realtime as far as the events in the video are concerned. For all we know the video could have been fast-forwarded to exaggerate the speed of the ferrari, and the timestamp added later.

This requires a bit of work, but do you notice this on frame 95?

That's a pedestrian traffic light, with a flashing green man to warn people not to start crossing if they haven't already done so. Mr. Blinky Green Man appears and disappears at a regular frequency of 1 Hz, accompanied by a countdown above on the red part of the traffic light, in seconds. Notice that the green light illuminates in frame 95, then goes out, then next appears in frame 121 (Figure 4²). This gives us a cycle duration of 120 - 95 = 25 frames, or exactly one second if the video frame rate is held constant at 25fps. Of course we also note that there remains an uncertainty of one frame, give or take, which gives us an error margin of 1/25 = 4%.

That the video is itself not doctored is harder to verify, but we'll just have to take it on faith for now. However, it's quite unlikely that the video could have been constructed to any particularly convincing degree of verisimilitude in the two days between the accident and its publication on YouTube. If it's fake, it's a very good, very detailed, and very realistic fake. But so far, the video seems legit.

The crash

Let's jump over to frames 506-513 (figures 5 and 6). Observe that the Ferrari crosses into the yellow box at some point between frames 506 and 507; similarly it impacts the taxi at some point between frames 512 and 513. Furthermore, in frame 506, the taxi is clearly shown to be a little more than halfway across the yellow box from the Ferrari (figure 7).

However, I couldn't find any accurate estimates about the width of the yellow box, so I did the next best thing and measured it on Google Earth. This wasn't helped by the fact that the box was actually trapezoidal, and not rectangular (figure 8). I obtained a measured value of $15.5 \pm 1.5~\mathrm{m}$.

As such we obtain the following:

Time taken for Ferrari to hit taxi after entering yellow box,

Distance that the Ferrari covered

Technically, this is a little bit of an underestimate, since the car is a bit more than halfway the yellow box away from the Ferrari, and less than halfway across the later half of the yellow box, but our uncertainty is already so horrible that I don't think it's worth quantifying that systematic error.

$\therefore$ average speed of the Ferrari in the box

This gives us an upper bound on the speed of the Ferrari at 147 km/h and a lower bound at 85 km/h, which is still holyshit fast, as lower bounds go.

Further considerations ³

Can we do better? Yes we can do better.

First, we need to consider the possiblity that the Ferrari was emergency braking⁴. I turned to the resident car enthusiast for data:

11:26:18PM Cheok Wei Ming: fact, the 599 GTO boasts an excellent 100 to 0 km/h braking distance of just 32.5 metres

This gives us a deceleration of $11.87~\mathrm{m~s^{-2}}$ at emergency braking, assuming a constant braking force from pure kinetic friction.

Using this to estimate the speed of the Ferrari, we arrive at $v_{\textrm{impact}} = (111 \pm 26.4\%)~\mathrm{km~h^{-1}}$, and $v_{\textrm{entering yellow box}} = (121 \pm 26.4 \%)~\mathrm{km~h^{-1}}$ to 3 s.f., which results in an upper bound on his cruising velocity at 153 km/h and a lower bound of 89 km/h, which is actually faster than if he'd been going at a constant speed throughout.⁵.

Watching the video we see that the Ferarri bounces off the taxi at an angle. This implies that it wasn't travelling at right angles to the taxi. Either it swerved, or the taxi swerved, or both, or their trajectories were angled against each other to begin with.

Using the ruler tool on Google Earth to measure their headings we realise that the effect of this is negligible ($1 - \sin \theta < 0.01$), and well within the error margin of our uncertainty.

Analysis

We have arrived at two sets of results, depending on whether or not Mr. Ma was emergency-braking. If he was, then he was probably violating Singapore's universal speed limit. If he wasn't, he was also probably violating Singapore's universal speed limit, but to a lesser extent. But then that would make us question how reckless he would have to be in order to not brake before ramming into another vehicle head-on.

An uncertainty of more than 25% is downright horrible. Note that although I gave my results to 3 significant figures (out of habit) it actually would have been more appropriate include fewer. Our result matches the official value of about 130 km/h, within uncertainty, but given the huge margin of error and inherent bias, that statement is almost vacuously true.

More than half of our fractional uncertainty resulted from the comparatively low frame rate of the video. I'd actually considered going to Bugis this afternoon to measure the traffic junction myself, but then I realised that this would still leave me with at least a 15% error margin, and decided it wasn't worth the effort.

Conclusion

So far we've looked mostly at lower bounds. However, note that in our estimation of the distance covered by the Ferrari, we've made two underestimates in our approximations which, though seemingly minor, not only introduce a nontrivial amount of fractional unceratainty, but also directly result in a systematically biased underestimate of the speed, too. So in truth, our lower bounds are probably quite a bit lower than they should be.

Even so, even our lowest lower bound of 85 km/h is stupidly fast for a vehicle travelling in the city. It's not just traffic-infraction-fast, it's criminally-insane-fast. It makes one wonder how many red lights Mr. Ma had beaten before his streak got terminated by an unexpected taxi.

By comparison, the EDMW analysis is a little off for several reasons:

• It assumes an incorrectly high frame rate, thereby decreasing the purported time interval between frames;

• It uses the white stop line as a measurement baseline, whereas said line is not actually sufficiently visible from the video to be used as an accurate fiducial marker owing to effects of perspective, physical line-of-sight obstructions and limited field of view, leading to an inflated distance baseline, and correspondingly an inflated velocity;

• And most importantly, in the immortal words of Walter Lewin, “any physical measurement is meaningless — absolutely meaningless! — without an associated measure of uncertainty.”

The first two objections I have to the EDMW analysis are also the case with the additional analyses that have subsequently popped up all over the place. This means that the vast majority of estimates you've seen being propagated by the online community are inflated, potentially deliberately.

However, I would surmise that it's proliferated across the internet because people will read, hear and, most importantly, repeat what they want to read and hear, even if it means using an inflated figure to somehow justify xenophobic outbursts. I do not approve of xenophobia, though I believe people have every right to be as wrong as they want to be. But I find it a little worrying that simple offhand remarks can be picked up by some people, thrown around the Internet sharing machine, and come out the other end politicised and either glorified and accepted as gospel truth or lambasted into oblivion. It says a lot about the inherent bias in both these factoids and the people that the machine is comprised of.

There, now that I've just made myself an exposed target for internet flaming by openly contradicting a Temasek Times Gospel Truth, I don't see much of a need to fuel the fire by adding any more of my own opinions into the mix. I hope you found this educational.