Sledding

I went sledding in Danehy Park in Cambridge recently, with my brother and some friends (including Mike, one of the other Boston-area people with a ball pit).  The snow was packed, icy, and awfully slick, and we were wondering just how fast the sleds were going at the bottom.

When you slide down something with no friction, your speed doesn’t depend on the path you take — just on how far you fall.  There are a number of simple equations derived from F=ma that are handy to memorize.  One of them gives the speed of an object after it’s fallen height “h” in Earth’s gravity:

In this equation, “h” is in meters and the answer is in m/s.  It’s actually 2*a*h, where a is the acceleration of gravity.  We round 9.8 m/s^2 up to 10.  (Other handy ones are that the time to fall that distance is sqrt(h/5) and the maximum range of a thrown projectile (45-degree angle) is v^2/10.)

This formula tells you that if your car nosedives off a 50-meter bridge (about double eastern US treetop height) you’ll be going about 30 m/s (interstate speed) when you hit the ground, making the crash the equivalent of hitting a concrete wall at highway speed.  It also tells you that if a (purely gravity-based) roller coaster’s highest cumulative drop, top to bottom, is 35 m (a typical large coaster), it can’t go faster than 26 m/s (which is roughly the old speed limit of US interstates).

I eyeballed the height of the hill to be about 11 meters, since I was about eye-level with the top windows of nearby three-story houses (Google Earth later verified this). So, the theoretical maximum sledding speed in Danehy park is sqrt(20*11), or about 15 m/s. In practice, because of friction, it will be lower (interesting note: the ratio of the vertical to horizontal distance the sled travels is roughly the coefficient of friction of the sled on the snow.)

Checking with our handy table, we see that 15 m/s is faster than the fastest sprinter, about the speed of a cat or rabbit (but — critically — slightly slower than a raptor), and not near highway speed.  We got the GPS from the car and did a few runs with it, recording the maximum speed each time.  It was a pretty reliable 10 m/s (11 if we pushed), which is a lot faster than running speed for everyone except Usain Bolt.

So, in every state except Wyoming, North Dakota, South Dakota, and Maine, there is a building high enough that Marty McFly could have taken the innards of his DeLorean up a freight elevator and acheived the required 88 mph (40 m/s) by jumping off the roof.  (Because of air resistance, I wouldn’t try it in NH/MT/ID/WV/AK/VT, either.) He just needs to leave a note at the bottom explaining that in 30 years they should set out a trampoline.

Edit: Sorry for the brief downtime today.  Also, to everyone posting that the GPS will only give horizontal speed and will underestimate speed on the slope: the sled reaches top speed near the bottom.  It only starts to decellerate when the grade of the slope is less than the sled’s coefficient of friction (plus a bit for air resistance), which seems to be less than 0.10 (people have trouble eyeballing slope grades, which are almost always shallower than you’d guess).  Since sin(x) ~= x for small x, this correction (1/cos(grade)) comes out to at most a percent or two in reality.  However, if the fastest part of the slope looked like the one in the second drawing, it would indeed be a big correction.

175 thoughts on “Sledding

  1. @Matt: Randall is correct that ones _vertical_ speed is path independent. However, ones horizontal speed and, thus, total velocity is dependent upon the shape of the hill, as Dr. Emmett Brown stated.

    @flo: Impressive find. Out of the German, I can pick out that it came to rest ~53 m horizontally & ~ 7 m vertically from where it left the ground. The interesting question now is what was the minimum velocity and corresponding slope of the ramp necessary to achieve this fantastic feat. For extra credit, don’t neglect air resistance.

    RE: Metric system: Math involving units is vastly more simple with the metric system. However, ten is not divisible by three, and in life in general, the need to divide by three is vastly greater than the need to divide by five. That said, the better solution (for this and other reasons) is to continue to use the metric system, but to change the _metric_ from ten to twelve. See http://www.dozenalsociety.org.uk (www.dozenal.org appears in to be in disrepair).

  2. I haven’t seen the Back to the Future movies in years, but doesn’t the DeLorean just travel through time (like in BTTF3 on the train tracks over the canyon) and not change its location or its direction? Wouldn’t jumping off a building get him going fast enough (discounting the power needed) only to travel him through time to crash right into the ground in the future?

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  4. I thought the wheels of the de Lorean had to be spinning at 88mph to switch the Flux Capacitor on. I’ve never understood how it evolved into a necessary component of time travel.

  5. @Matt: Randal is correct that the vertical component of velocity is independent of the path. However, the horizontal component and, thus, the total velocity _is_ dependent upon the path. The ground exerts a force on the sled normal to the slope, with the vertical component of the normal vector equal to the force of gravity (otherwise you would displace the ground). Thus, the steeper the slope, the larger the horizontal component of the normal vector and greater horizontal acceleration. The Brachistochrone curve is the quickest path between two points and therefore has the greatest speed. Take a quick look at the shape and you’ll see why the maximum velocity is at the bottom, not the end of the curve.

    @flo: Impressive find. From the German I was able to glean that the car flew some 53m and ended up 7m higher than where it started. The interesting physics problem here is to calculate the minimum velocity and corresponding slope of the ramp needed to achieve that flight. For the ambitious, don’t ignore wind resistance.

    RE: metric vs ‘imperial’ system: The place where the metric system shines is in doing math with units. However, the imperial system has greater practical usability, first and foremost because it allows clean division by three, which occurs vastly more often than the need to divide by five. Thus, the best solution is to combine the strengths of each by using the metric system, but changing the metric from 10 to 12. Search for dozenalism for more info. There’s a very practical reason many things are sold by the gross, 12^2.

  6. 88mph is never explained past “activating the flux capacitor”.

    The icy of the surface of the DeLorean is probably the one thing that the movies got right about the mechanics of time travel. One concept on how time travel would work is to capture two quantum singularities connected by a worm hole, send one of the singularities rocketing around space at nearly the speed of light, and bringing it back home after a set amount of time. The singularity that you kept at home would have aged more than the one you sent on a cruise. If you were to then open the singularities into some sort of tunnel, then you could walk through one singularity and end up coming out the other end, having stepped back through time. Never mind how you’re supposed to capture a singularity (let alone two, especially two connected through time and space), or how you’re supposed to “open” one (apparently some sort of negatively charged gravitons are the way, despite neither knowing if the graviton exists or if it even has a charge).

    However, given that the singularity is an infinitesimal point, there would be no matter inside this worm hole, and therefore no heat. IIRC, the vehicle spends a significant “time” in the intermediate space between spaces (I believe this is seen in the second movie), and they have to leave before too long or the occupants will freeze (I seem to recall Marty being quite cold at one point). Once the vehicle returned to normal space, the moisture in the air would frost on the surface of the car, like taking a glass mug out of the freezer to fill with ice cream and root beer.

    Mmmm, root beer floats. Definitely not enough of those in my life.

  7. I wonder if the Delorean would time travel properly if it was doing 88mph on a treadmill? :)

  8. @Brandon, that’s what the note about the trampoline is for. Unfortunately the plan in post only works for forward jumps and that only if there’s someone you can trust who will ensure that the safety net gets set up.

    An alternative way to go about this would be to set up a safety net at the bottom of a building of appropriate height right now. If you did that you could jump back to the point in time where you set it up from any point in the future.

    Of course if you don’t immediately see your future self appear falling from the sky then it means you never have the opportunity and motivation to follow through on this plan or time travel is based on some other plot maguffin than velocity.

  9. @Jazzman

    You need to set up the treadmill in a wind tunnel to get the appropriate intertial frame. How else are you going to down the timeline faster than the timeline?

  10. I just realized how much this helped me in preparing for my physics midterm tomorrow – especially the shortened formulas. =D Thanks!

  11. Ah a fantastic new blog entry that just goes to show how raptors* are in fact > sled + gravity.
    Also shows why Randall should update his blog more often.

    @Spherical ‘your mother’ on snow joke: Assuming she isn’t sliding, instead rolling, you would need the coefficient of static friction.

    Captcha: rau upsilon

    *assuming pleisosaurs for stability on snowy surfaces.

  12. @Ben not quite. Randall has it exactly right (since he said speed instead of velocity, he didn’t even need the word “magnitude”), the magnitude of velocity is dependent only on the change in height, ignoring friction. Simplified, this is because conservation of energy gives the exact amount of energy that has been converted from potential to kinetic, so the velocity magnitude can be determined from the kinetic energy equation by plugging in the change in energy of g * delta h, as all forces considered are conservative and no other object gains energy (and none of it is converted to angular momentum, hopefully).

    Thought experiment for proof: If the vertical component of velocity of an object was determined only by height fallen, at the end of any arbitrary ramp, an object would hit the ground with the same impulse as dropping it from the initial height. If the ramp curves to level with the ground before it ends, this is clearly an absurd assertion, as there will be no remaining vertical component to the velocity.

    Since the ramp is exerting some upward force on the object descending it, the object cannot be vertically negatively accelerating as fast as if it were in free fall, and with the distance an object travels under constant acceleration in a direction being .5*a*t^2 and the velocity in that direction being a*t, you can show that since the vertical acceleration has a smaller magnitude, so must the vertical velocity at any change in height (set the velocities equal for the two values of acceleration to find the difference in time to reach a velocity, plug into appropriate distance equation).

    If the vertical component of the normal force on the sled of the slope is equal to the weight of the sled, the sled will not accelerate downwards. The reason the ground does not move is that the sled is not exerting it’s full weight on the slope, because it is accelerating vertically in the downward direction.

    The reason the Brachistochrome curve is the fastest path under gravity is because the magnitude of the velocity is larger than a straight line path while it is below the endpoint. What Randall, and everyone else is talking about, is the speed after leaving the ramp, which even for this curve, is still sqrt(19.6*h), as h is the difference in height of the endpoints. While the curve has greater average and peak speeds than a straight ramp connecting the endpoints, if you want to compare peak speeds in our context, our ramp must reach the maximum change in height of the curve.

    Sorry for the long post, I’m an amateur physics nut.

  13. There’s no real reason why the GPS unit wouldn’t be able to tell your speed even if you’re moving straight up and down. It requires at least 3 satellites to triangulate your position on the globe – and assuming you’re on this side of the GPS satellites, you don’t need a 4th one to find how high you are from the sea level. In fact, the GPS unit actually shows how high you are. Only if it’s really poorly programmed, it won’t take height into account when calculating your speed.

  14. @ Sean T. McBeth,

    I always thought that in space you would lose relatively little heat. Since there’s no air or solid surfaces touching you, you can’t lose any by conduction or convection, only radiation, which would be fairly slow. Regardless, I don’t think there’s any water to ice the surface over. Mind you, I’m not a physics major or engineer, nor have I even sen back to the future.

  15. @Matt: Radiation is relatively minor on Earth because most things are at the same temperature, so are relatively close to an equilibrium – you los X amount of heat energy radiatively per second, but gain Y by radiation from your surroundings, and the numbers are quite close due to the small temperature differences. In space, neglecting nearby stars or planetary destruction lasers, you’re in a thermal bath of cosmic microwave background at 4K – this is a rather large temperature difference, so heat loss would be rather faster.

  16. McFly could jump off the top of the Crazy Horse Memorial (or the mountain that will be the memorial eventually) in South Dakota.

    The Crazy Horse mountain is approximately 172 meters high (sourcing Wikipedia), which is considerably more than the needed 76 meters to achieve 88mph (and is tall enough to allow for plenty of leeway with air resistance).

  17. @Storm: And I thought that it wouldn’t have been possible to pull ‘your mom’ jokes here because of the nature of potential and kinetic energies:
    Since E_pot=E_kin with E_pot=mgh and E_kin=(1/2)mv^2, we get mgh=(1/2)mv^2 and we can divide by m on both sides of the equation, thus cancelling out m and obtaining gh=(1/2)v^2, or, when solved for v, as stated above, v^2=2gh and v=sqrt(2gh), approximated to v=sqrt(20gh).

    When I was a kid, it really seemed completely counter-intuitive, but really, your mom ain’t getting no faster than my mom when the sled is at the end of the slope.

    Maybe, though, someone would like to redo this calculation when friction effects (sled/snow and mom/air) are involved and see where the physics go.

  18. IMO the point in sledding isn’t the top speed, I do it just for the thrill and fun. You can’t get the same feelings in a car at highway speeds when you compare it to sliding down the hill in a czechoslovakian nuclear fallout suit (without the sled, obviously). Wintery greetings from Finland, it’s -20C here and lots of snow…

  19. This is perfect. I just finished a runthrough of projectile motion in physics AND i watched all three back to the futures last weekend. good timing.

  20. There’s at least one gadget I know of that records your 3D position based on GPS. (My dad has that for his mountain biking expeditions.) That should be good enough for recording the actual speed.

  21. For some reason, I now have an image of Calvin and Hobbes carrying on a similar conversation about physics as they whiz down a hill…

  22. @Matt: also, there doesn’t need to be any water in the vacuum for the ice to form on the car. If the surface of the car gets cold enough, it will frost over almost immediately when it reenters the atmosphere, which always has at least a little humidity. Like I said, think about taking a glass mug out of a freezer before you fill it with root beer for your root beer float. No, it won’t make plates of ice that crack and fall off, but it will fog.

  23. Actually many of the states you mentioned (especially dakotas and wyoming) have radio towers that are plenty tall enough. I believe the tallest structure in the world is currently a radio tower in N Dakota. Based on my experience climbing them, I would say Marty would have a tough time lugging the flux capacitor and apparatus to the top, especially with the shaky hands and whatnot that he’s got now.

  24. “(Because of air resistance, I wouldn?t try it in NH/MT/ID/WV/AK/VT, either.)”

    I don’t quite understand this parenthetical bit; is the air density more in these states? Otherwise a great post with physics and whimsy… I wonder if Calvin and Hobbes ever gives an indication of how high their hills were.

  25. After today’s comic, I would just like it know, $5 wrenches can be bought from Auto-Zone, probably the LAST place most of us would look for ANYTHING.

  26. @everyone about the GPS:
    GPS is actually relatively poor at determining altitude (of moving objects), and even our best DTED Level 2 calculations have “a post spacing of 1 arcsecond (ca. 30 meters).” (http://en.wikipedia.org/wiki/DTED)
    From http://en.wikipedia.org/wiki/GPS#Possible_Sources_of_Interference and http://en.wikipedia.org/wiki/GPS#Error_sources_and_analysis, you see that there are a lot of errors involved, not the least of which is that the earth isn’t quite close to a perfect sphere. Most of this is usually ignored, but the change in altitude is an averaged and ill trusted GPS feature for things on the move. Thus, most fit the data to DTED maps built into their maps of earth.
    All of this is why most GPS programs do not take height into consideration.
    The super accurate GPS’s require you to stay still for a little while while it averages things.
    Anyway, in the end it doesn’t matter, because Randall’s explanation makes all this moot.

    My $0.04 (inflation)

  27. @Alec:
    As stated above, an object must freefall for approximately 76 meters to reach 88 miles per hour 39 m/s). The reason Randall wouldn’t try time-travelling in those states because those states’ tallest buildings are *just barely* taller than 76m.

    For example, the Conoco-Phillips Tower here in Alaska is approx. 90m high (see link in original post.) According to my calculations, a jumper would reach just 93 miles per hour (42 m/s) before hitting the ground, assuming no air resistance. That’s not nearly large enough of a margin of safety.

  28. Whistle while you work, Randall is a berk!
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  29. GPS units utilize “triangulation” in three dimensions. (This is how they are able to tell altitude.) Remember in grade school when you did the lab on triangulation of earthquakes based on three points? Three radii define one point in two dimensions. (In other words, n+1 points are needed to triangulate in n dimensions.) That means that 4 points are needed to triangulate in 3 dimensions.

    Now, your GPS unit routinely gets at least 6 satellites, even more on a clear day away from trees, buildings, etc. (The unit in my car tells me the exact amount of satellites it gets at any given time. I’ve seen as many as 11 satellites at once.) More satellites above 4 simply gives you a finer resolution of position. (As a side note, your GPS will function with only 3 satellites, but it will not tell you altitude.)

    That GPS unit in my car tells me my speed at any given time. It ALWAYS matches the speed shown on my speedometer. The speedometer obviously shows the speed the wheels are moving at. If the GPS only showed lateral (and not vertical) speed, when driving up a steep hill or mountain, the GPS would show a lower speed than my speedometer. This is not the case.

    The GPS calculates a position in three dimensions and uses the corresponding change in all three dimensions to calculate your speed.

  30. danahy park is fine, but try kingsley park, right on fresh pond next time: the path that cuts across the sledding slope partway (1/3? ish.) down makes a natural spot for a jump if there’s even a medium-heavy snowfall, and the closest-to-the-pond end of the slope is good and steep.

  31. Nerd note: Marty would NOT want to do this.

    First of all, the DeLorean was picked (at least partially) because of its stainless steel construction, which, when the special effects of the car in transit are considered, indicate that there is some serious electrical energy being thrust through the outer shell of the car…energy that would likely kill Marty.

    Second, time travel in Doc Brown’s machine involves a lot of heat loss (this is sadly ignored in the sequels). The car comes out covered in ice (I assume it’s just a lot of flash-frozen ambient moisture, as opposed to the machine creating dry ice or something), which would be…unpleasant, to say the least.

    Third, speed was not a problem in the first or second movies. Plutonium and Biff were. Speed was only an issue in the third movie, set in the Old West, before most of those buildings were constructed (besides, imagine trying to take the DeLorean cross-country in the 1880s). Doc dismisses the sled option because of the resistance issue (“We’d never find a smooth enough surface!”).

    Though, in retrospect, it would have been a lot better for the timeline to push the car off a cliff and hope for the best, rather than opening a blacksmith’s shop, saving a folk hero from the death that made her a folk hero, hijacking a train, and kidnapping the aforementioned folk hero.

  32. If he used a handy Aperture Science Handheld Portal Device on the way down, Marty should be ok. Maybe.

  33. In all honesty, I didn’t read the post. Although, judging by your awesomeness, it was most likely hilarious.

    Formalities aside, I just realized that you commented on my small-time blog; this for me, is christmas in february. The fact that you even found it is a great token of respect for me, and I sincerely wanted you to know how honored I was when I found out. And yes, of course I read your work. Witty, sublime and altogether fun.

    Kudos, on the humorous prowess and extravagant success. You are my web-comic makin’ guy hero and I sorely wish I had your talents.

  34. Ahoy there, fellow web-comic sailor!

    I only just realized a few days ago that you commented on my humble blog. Honestly, I wanted you to know how big a deal that is for me; you, as far as I can tell, are the almighty king of hilarity on the inter-webs and receiving a comment from you is like christmas in february!

    Yes, I love your work. It is my inspiration, and I only hope that some day, I’ll be able to write web-comics with your wit, charm, subtlety and sophistication. Kudos on the web-comic prowess.

  35. Randall, if you’re reading this, I have a question totally unrelated to this blog post:
    Do you play any muhmorpuhguhs?

  36. As for the current comic “Boyfriend”, the character at the end says “Your math is irrefutable”. Please, statistics isn’t real math. It’s to mathematics what sociology is to social science, people do it because they can’t do the real stuff. “Rithmatic” requires rigor, repeatability, and reliability (alliteration ftw). Statistics is pseudo-mathematical hand waivery. Reliance on statistics gets you all kinds of blown-up things, like shuttle crafts and housing markets.

  37. We just got to the laws of motion in my physics class.
    This was very cool to read. Will use physics when next time i go sledding.

  38. Wouldn’t it be easier to just take the freight elevator to the top floor, then cut the cable? Then thirty years hence, have someone take the elevator to the top floor and leave a bungey cord hanging from the top?
    And a trampoline at the bottom? I’m just trying to be practical.

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