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[WCYDWT] Pure Performance

By • December 10, 2010 • 65 Comments

Can I get forty comments on this video? The rest of this post (which I’ll update after we cross that threshold) depends entirely on the modal answer to the question, “what question perplexes you about this video?” Ask your own question before you look at the others.

BTW:

My advance prediction for the responses was a million-way tie for third place with these two responses taking the top two spots:

  1. How fast was the car moving?
  2. How many frames were on the wall?

I couldn’t guess which would win and, in terms of our mathematical objective, it doesn’t matter. Both questions are tightly interwoven. (Many commenters, in fact, asked both questions.) It mattered to the production company, though. One is its premise. The other is its conclusion.

The results, through Yaacov’s comment:

Sarah Cannon: “How fast is the car driving?” +25 related
mirjam: “how many pictures?” +16 related
Brent Logan: “How did the car jump? There wasn’t even a ramp.” +7 related
Other questions. +7

The behind-the-scenes video is less helpful than you’d imagine so I spammed basically every production company that had anything to do with the ad, asking them for information. Everyone was extremely helpful (why is this such a revelation?) and eventually my query was routed to Peter McAuley, the visual effects supervisor, who dropped a pile of knowledge on me, which I’ve uploaded here and which he cross-posted in the fourth comment here.

Important questions your students may have to reckon with.

  1. How does this effect even work? More to the point, how does film work. The camera on the outside of the car is taking photos very quickly – at a rate of 24 frames per seconds. When they’re played back, our brains interpolate the rest of the motion and it looks smooth.
  2. Did the production company decide on the size of the frames and then figure out the speed or did they know the speed they wanted and then determine the frames? The production company figured the faster the speed, the more exciting the effect. At a certain speed, though, even the best digital sensor produced a “rolling shutter” effect (excellent animated explanation here) so they went with film, which looked good up to 40 kph. Which, bingo. That figure is crucial to every question.
  3. How long is the actual ad? This is the extended cut of a TV spot that ran for thirty seconds.

Start with discussion, brainstorming, estimation.

What’s your guess? How many frames do you think you saw fly by? How wide do you think each picture had to be? Give a number you know is too high / too low.

Gradually apply the mathematical framework.

Ideally, either on their own or throughout the class discussion, your students will realize they need a) the length of the ad [30 seconds], b) the speed of the car [40 kph], and c) the shutter speed of the camera [24 frames per second].

From there, the dimensional analysis is up to you and your students.

Show the answer.

Download the goods.

Zipped archive [135.2 MB] containing:

  • the extended commercial,
  • the behind-the-scenes video,
  • a PDF of the Peter McAuley’s e-mail.
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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

65 Comments

  5. John Golden

    December 10, 2010 - 9:54 am -

    Wow. I love the measurements, 2 degree shutter (what does that mean?), 72 frames, 24 fps, 5 ft 35mm… which suggests 3 sec of film, why put that in? Lot more than 3 sec of animation (how long?) That is a traditional animation speed, 24 fps (pretty high quality) how does that compare to digital video or high resolution digital video. But my main question… what does the camera have to do with it? It’s mounted to the car for the driver shot. Rewatched: now I think the measurements are just for the lead, a countdown to the film starting. How fast would the car have to go to jump the chasm? How did it gain altitude?

  6. Debbie

    December 10, 2010 - 9:55 am -

    What factors needed to be worked out for the cartoon to work? (Speed of car, distance between images..?).

  8. Datin

    December 10, 2010 - 10:04 am -

    I’m a film major. I wonder how you DO this. Given the frame rate of the camera, and an estimate of the frame size, I should be able to create a formula that tells me the distance required between frames for any given speed.

    My next question is how do you get in sync? I think it requires a human touch to make the tiny adjustments needed to put the image in the center of the frame at speed.

  9. Scott

    December 10, 2010 - 10:10 am -

    Its an amazingly great idea, completely ruined by the fact that people won’t “get” it. Too bad. We need to understand that what we are seeing is the view from the driver’s window. We should have seen him pull up to the wall, both from his camera’s point of view, and from a third point that shows the camera. The effect is the cool thing – not the “surprise” at the end of how they did it.

  11. mirjam

    December 10, 2010 - 10:13 am -

    1. how fast does the car have to be going to jump that way?
    2. was there a ramp or did it jump without it? is it possible?
    3. how many pictures?
    4. how long is the wall?

  12. Bryan McDonald

    December 10, 2010 - 10:25 am -

    For those of us who work in schools that find it necessary to block all video, it would help if you would post the direct download link for the video like you do on so many of your other WCYDWT posts.

  13. Mike Poteet

    December 10, 2010 - 10:33 am -

    Would this really work? I thought the principle behind animation was that numerous but discrete images were being photographed, one after the other, and then run all together so that the mind creates the illusion of motion. If you simply run a camera in a continuous motion past a series of discrete images (i.e., without opening and closing the shutter), wouldn’t the resultant film just be a big long blur? Or am I missing something?

    (As requested, I posted my question first before reading the others, so apologies if I repeated someone else’s comments.)

  14. Dave

    December 10, 2010 - 10:37 am -

    How did they sync properly after acceleration? That is, how is the shutter catching the still images on the paper at the correct instant, when they are centered in the camera’s view? (I’m assuming that acceleration is unpredictable and that the camera isn’t adjusted during the run. Seems like there might be some movie magic here, either zooming into a centered frame in post editing, or this is the best run out of many tries.)

    They provide a lot of the data up front, so a lot of the math questions that would normally make me curious seem pretty straight-forward. It’s like a word problem, where you know all the numbers you need are in the paragraph, so you just have to grab the right ones…yawn.

  15. Benoit

    December 10, 2010 - 10:51 am -

    Is it really possible for a car to jump like the cartoon one? I doubt it. Why does the spider fall in the hole?

  16. Andrew

    December 10, 2010 - 10:52 am -

    How fast is the real car?

    On a second viewing, I found myself looking at the blurred side images once the car reaches sync velocity, and wondering about the width of the animation frames.

  17. Matt

    December 10, 2010 - 10:53 am -

    How fast is the car driving? It seems you would be able to calculate this and would need to use the information given at the front or use some estimates of the number of frames and their lengths.

  18. Ed Hitchcock

    December 10, 2010 - 10:54 am -

    In the animation, how does the car jump without a ramp? In the real world, how fast must the shutter speed be for the individual frames not to be blurred?
    We know the frame rate, so if we knew the frame size or speed or relative distances, we could find out missing values. We can, however, estimate:
    From the end of the video, it looks like the distance to the wall from the camera is ~2x the camera-helmet distance. From near the end, the frame size appears roughly equal to the helmet size, so at 2x the distance, it is 2x the size. Frame size would then be ~ 60 cm. At 24 fps, the speed would be ~52 km/h (32.5 mph). Not that fast, really.
    So then, to answer my initial question, if the lines in the drawing are 3-5mm thick, the shutter has to be fast enough that the car is not moving more than that distance during a single exposure. At 52 km/h, that would mean th eactual shutter speed is about 1/3000 s.
    So my actual questions are:
    Is this legitimate, and if so, how many takes did it require? Certainly the final shot shows the run at dusk, so they were probably at it all day. The light at dusk would most likely be too dim for 1/3000s shutter speed.

  20. Devan

    December 10, 2010 - 11:03 am -

    How many individual frames are there?
    How fast is the car driving?
    How long is the wall?
    How many “blank” frames do you need while the car gets up to speed?

  21. Angie B

    December 10, 2010 - 11:21 am -

    How far across was the ravine?
    How fast would the car have to be going to jump it?
    The ravine looks like it’s straight across/ flat, so is jumping it really possible? Wouldn’t the jump go in a parabola?
    The beginning number frames: What did they all mean? It looked like a seconds count down until filming, was it?

  24. Kris Kramer

    December 10, 2010 - 11:59 am -

    1. How fast is the car moving?
    2. How long of a strip does he drive?
    3. How many “cartoon” frames were needed?

  25. Simon L

    December 10, 2010 - 12:00 pm -

    The first question I am keen to answer is:
    How many images/drawings/pages of the ‘flip book’ are there?
    Other questions that interest me include:
    How fast is the car travelling?
    What is it’s acceleration?

  26. keninwa

    December 10, 2010 - 12:03 pm -

    3 questions come to mind. Here they are in the order I thought of them:

    1) To what do the dials at the beginning refer?
    2) What makes the animated car “hop” the ravine?
    3) What is the cruising speed of the physical car?

  27. Chris Sears

    December 10, 2010 - 12:08 pm -

    My first question is “How does the car stay at a constant speed?” I drive an Outback, and my last car was a Legacy. The speedometers were always fast by three to five miles an hour.

    My second question is “How long is the paper?”

    My final question is “Can we use the flicker in the first few frames to predict the acceleration of the car?”

    I actually wrote a blog post about the connection between the Subaru logo and Halloween, if anyone is interested.

  28. Emily

    December 10, 2010 - 12:15 pm -

    How fast is the car going? (Using the frame rate information at the start of the video, I think… not sure if there’s enough there.) Although if I cheat and look at the answer above me as I type this, I wonder about the acceleration too.

  29. Matt

    December 10, 2010 - 12:31 pm -

    How fast is the car moving?

    What’s a two-degree shutter?

    Assuming the real car is on a racetrack, did the production crew have to deal with the curvature of the track? (Which is two questions, I suppose: Was the “film” longer than the straight part of a track? And would it even matter–if the film wrapped around a turn with similar curvature to a racetrack, would it distort the image?)

    Speaking of distorting the image and the speed of the car, how fast would the car have to move for a kind of visual Doppler effect to be noticeable? At some speed, the frames of the film would have to be stretched horizontally so as not to appear squished on the recording. The way I’m phrasing it makes it seem as though it would take relativistic speeds, but my gut says it might happen a at much lower velocity.

    And above all, can we replicate this?

  31. Ben T

    December 10, 2010 - 12:45 pm -

    The question I’m really curious about is: How fast is the car going once it gets up to speed? I don’t know if I’d say that question perplexes me, though; it seems like it would be straightforward to figure it out. Things others have suggested are much more perplexing and probably more interesting, but I feel like the speed is important to how cool this video is.

  32. Andrew

    December 10, 2010 - 1:03 pm -

    Sadly, I just showed it to a group of 7th graders in a study hall, and they were underwhelmed. Many didn’t understand the mechanics of animation or filmstrips, and expressed confusion that “all the pictures on the wall are the same.”

  34. Z. Shiner

    December 10, 2010 - 1:39 pm -

    How fast is the car moving?

    How many pictures did it take to make the shot?

    Then I had to look at the starting numbers again and wondered what they meant… 3 seconds at 24 frames per second (the middle number) means that there would be 72 frames total (the top left number). What’s it all mean, jellybean?

    What’s 5 feet at 35mm? Why does it go to 4?

  35. Audrey

    December 10, 2010 - 1:42 pm -

    How fast is the car going when it is up to speed?

    I’m not sure that is the most interesting question at all but it was the first one I thought of. I had to watch the video a few more times before I started thinking of any of the other questions I had (all of which are included in the other responses to this post).

  36. josh g.

    December 10, 2010 - 1:54 pm -

    How many drawings?

    How long is that friggin’ paper?

    How fast was he going?

    (In order that I thought of them, with the last two appearing right at the end. I think “How fast?” is the one that has me the most hooked.)

    Now to actually scroll up and read what other people thought of.

  37. Ben Blum-Smith

    December 10, 2010 - 2:10 pm -

    1st time thru:

    WTF does the camera and the car have to do w/ the animated car?

    2nd time thru:

    Oh.

    How many frames? / How long per frame? / How are the frames spaced? / How fast? And, is the camera catching every frame?

  38. Luke

    December 10, 2010 - 2:13 pm -

    2 questions:

    How smoothly does the car have to drive in order for the animation to work?

    Why does the animation only show up in one location, if the pictures are going across the entire field of the camera?

  39. Tom

    December 10, 2010 - 2:13 pm -

    What happens if the car goes twice as fast? Half as fast?

    And as with any video that seems this slick — what do the original takes look like? How many takes did it take? Did they cheat?

  40. Tim

    December 10, 2010 - 2:50 pm -

    How fast is the car going? And how little do I need to know to determine that?

    I think frame rate and length of frame are enough…

  43. David

    December 10, 2010 - 3:21 pm -

    Honestly, the first question that popped into my head was where’s the behind the scenes video.

    And then I went and found it.

  45. Dave Gale

    December 10, 2010 - 11:55 pm -

    well, i thought:
    How fast?
    How long is the paper?
    How long is the clip?
    How many people drew the pics?
    How long would that have taken?
    How dò you keep the car at a constant speed?
    How many takes did it take?
    What difference would it make to the speed if the pictures were twice as wide?

    There are Probably some more we can think of.
    Dave

  46. Dave Gale

    December 11, 2010 - 3:16 am -

    Although, the question that perplexes me the most is:

    Will the effort and money that’s been spent on this directly result in improved numbers of people buying the car?

    (and how many cars need to be sold to break even?)

  47. Ed Hitchcock

    December 11, 2010 - 6:30 am -

    Okay, so I was off by a bit in my estimates, but not too bad.
    It would still need a shutter speed of ~/2000 to not blur the images. They must have one seriously fast f ratio on the lens.

  48. Mr. K

    December 11, 2010 - 9:44 am -

    I’m late, but my big question is “how do you get it to synch so the frames are centered”. The camera rolling ahead of time makes it difficult to synch once you’re up to speed…

  50. Shirley

    December 11, 2010 - 7:15 pm -

    Something that always amazes my children (and even many adults in the room) is the visual effect that makes the wheels appear as if they are rotating backward instead of forward. The discussion of frames/second and speed of car et al above is a good lead-in to exploring that phenomena.

  51. Jon

    December 13, 2010 - 4:06 am -

    Is the animation sequence actually recorded using the motion of the car? What actual proof is there that this is done using this method? The car driving by the images may work based upon the film speed. My bet is that they created a computer image. Where are the driver’s arms and the steering wheel in the shot?

  52. Olga

    December 13, 2010 - 5:57 am -

    An extension project, maybe: Tunnels in a lot of cities are sooooooo boring! and UGLY. I often pondered if there was a way to create an animation-like something on the walls of the tunnels. This answers several questions – first that it’s possible, second that the speed is reasonable (40 mph) for city driving, and that it would actually look cool. While this may create a distraction for the driver, and this is a serious issue, it will be so much fun for passengers!

  57. Charles Breiling

    December 15, 2010 - 5:23 pm -

    My first question was only about the cartoon reality. Sure it’s possible to “jump” the chasm with no ramp, the far side just needs to be at a slightly lower elevation, provided sufficient speed of the car. But if the car and the spider are going the same speed, why is the spider “more affected” by gravity? I guess the folks at ACME Robot Spiders could answer…

  58. Karl M

    March 7, 2011 - 1:08 pm -

    How fast does a car need to go for it to appear that the wheels are turning backwards? The refresh rate of our eyes, or whatever the human version is. Rotations of wheel per second, or seconds per rotation at given speeds. Loads on that little beauty!

  59. Karl M

    March 7, 2011 - 1:10 pm -

    Sorry, no edit function, with big wheels, how fast would a car need to travel for backwards motion? And a tractor?

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