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.

171 thoughts on “Sledding

  1. Oh man. I can never read those kinds of sentences. My primitive brain can’t process past the “rong” and “u” and “4.” :( IZ NOT GRAMMERIC.

    Anyway. This is the first blog I’ve had the balls to post on… a big step for Miss Antisocial here. ^^ I love this comic, dude, FYI. Always seems to be relevant in conversation.

    WHATEVERYOU’VEHEARDTHISAMILLIONTIMES… ><

    You could have gotten one of those speed-tracking guns policemen use. If you steal one, you’d be sticking it to the man AND doing science!

    On a side note: When those GPS voices try and steer you into a ditch, as mine does on occasion, do they not seem to have a slight maniacal glee in their voices…? It gives me the chills. Just one short step until the machines will have utter power over us… http://xkcd.com/251/

  2. Well, here in Florida we can’t even get close to physics land. :(
    Can’t find a near-frictionless surface created naturally in nature, and unfortunately we have lots of air resistance here (I’m going to take a stab in the dark and say that humid air provides more resistance than dry air because the water being held by the air is more dense (why our oceans/lakes are full of water and not air) ? = m/v and since momentum p=mv and since stuff wants to stay at rest if its at rest (Newton’s first law) water in air will have more inertia than regular dry air = more air resistance.

    For long drawn-out reasons like this, is why we complain to our AP Fizzix teacher that none of the stuff we learn is applicable to real life.

    [Hey, I never realized how ?=m/v and p=mv looked so dang simliar!]

    Other than that, I wanted to mention to people that in your comic “Base system” that the bit of binary says “base 2″

    Clever Randall. Shrewd, indeed.
    However, its placement leaves this virgin teen water polo (not baseball) player stumped. [I have no placement on your field]

  3. Your comic randomizer has a slight error- it took me to Comic 404, which doesn’t exist.

  4. I don’t suppose you heard about someone hacking in to the road signs in Indiana, Illinois, and Texas warning travelers about zombie and raptor attacks?

  5. Definitely partial to “Linux (or BSD :) would…”.

    Also, I’ll insert a space between a URL and any adjacent punctuation, to keep braindead clients on the other end from treating my punctuation as part of the URL. (Trillian!)

  6. Would the sled gain or loose speed by falling through thin ice? I assume that it would slow down once it was actually submerged, but on ice the friction would be lower. No slope though, so it makes sense that the sled would only decelerate at a slower pace. But for the sixteenth of a second it would take to fall through the iced section and hit the water below, the sled would have a momentary boost in total velocity (again assuming that the ice is perfectly horizontal.) Plus, in a really shallow pond (That is, the depth of the water is less than the height of the sled) would the sled go faster, even in liquid water? Hydroplaning?

  7. i have actually contemplated this dilemna quite a bit
    instead of making the smile out of a colon and parenthesis, make it out of a colon and a capital D

    Linux (or BSD :D)

    i have yet to figure out what to do for frowny faces. for now, don’t say anything negative. or use the extreme power of sarcasm.

  8. I agree with Kim & Xander, the best solution is to avoid ending in a parenthesis. ^_^, >_>, :D and the like all work very well while enclosed.

    If you must end with a parenthesis, use the emoticon after the sentence (like so). :) Most of the time, the smiley is distinct enough that the beginning of the next sentence is clear.

  9. drop the parentheses entirely. It should look like this: Linux [or BSD :)] would…

    there you have it!

  10. i would expect a GPS to calculate speed going at an angle. my GPS can do altitude so should be able to calculate the speed in 3-space

  11. Randall is a faggot.
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  12. /I’ve never understood how [88mph] evolved into a necessary component of time travel./

    Overthought answer: The temporal equivalent of static friction, the relativistic time-dilation (~1.0000000000000087) (relative to your environment) represents the minimum amount necessary to become “unstuck” in time. (Like how you can stand still on slippery ice with solid footing, but once you START to slide, it becomes much ‘slipperier.’ Static friction.)

    Simple answer: The Delorean apparently rips and hole and travels through. It has to be traveling /at least/ 129 feet per second in order for its 14′ frame to make it through that hole before it collapses. (in ~1/10th of a second.)

    Moderately-thought-out-answer: The simple answer can’t be correct– the train can get through at 88 MPH. But all time-trips seems to “occur” in the same amount of time for the observer– 88mph may be a trade-off; the minimum time for the time vehicle to transition the blast-freezer-cold safely.

    Rational answer: It’s a flux capacitor. Whatever “flux” it stores (relativistic?) alters with velocity, and 88MPH is the “magic” point where the generation-and-storage rate exceeds the bleed-off rate.

    Best answer: That’s the speed it goes. It’s like asking why 7m/s is escape velocity. It just IS.

  13. “It’s like asking why 7m/s is escape velocity. It just IS.”

    But it’s not. Ignoring the order of magnitude mistake in your units, escape velocity from the surface is 11.180 km/s. Orbital velocity (and by that I mean circular orbits) at the altitude of the ISS is ~7.7 km/s, so perhaps you meant orbital velocity.

  14. Regarding your smiley conundrum, I think I’ve figured out the answer (all you have to do is to be left-handed about it (: ).

  15. @Yourself- in this case, m/s is miles per second, not meters. 11.180 km/s is 6.95 miles/sec. Gotta go beyond the metric system sometimes, son.

  16. you know i just learned this in high school physics.

    its derived from potential and kinetic energy equations.

    I love you xkcd

  17. I love physics!

    I didn’t realize how fun my high school physics class can be (planning on taking AP-C next year)

    I have to blame you Randall for making me get into physics.

    Thanks for the aweomse webcomic!

    p.s.(first time poster)

  18. “@Yourself- in this case, m/s is miles per second, not meters. 11.180 km/s is 6.95 miles/sec. Gotta go beyond the metric system sometimes, son.”

    Using “m” as an abbreviation for mile is blasphemy of the highest order. I can work in either system (I was brought up on the English system even), but the only time “m” means mile is mph or mpg.

  19. >but the only time “m” means mile is mph or mpg.

    And since the DeLorean goes 88MPH, it’s safe to assume English units are used throughout, yes? I mean– it’s not going 141.6KPH, is it?

    I wonder if the DeLorean EXPLOITS the universe’s confusion over the unit conversion? Sneaking out of the timestream as it bogs down, trying to properly calculate the dynamics.

    It would explain why the tire friction sim continues for several seconds after the machine is already gone!

  20. An interesting question is what shape should the hill be, so that you get to the bottom in the shortest time. If it’s a constant slope you take too long speeding up. If it’s almost vertical at the start then flat you take too long not moving forwards much.

  21. @Sidd
    your question has been answered in most Calculus books that address parametric equations. The answer is a cycloid (inverted). This is the curve that a point on a wheel draws as the wheel rolls without slipping. I don’t really want to find the equations and copy them here. But like I said, you can find them in a Calc book. And probably Wikipedia…

  22. nick, i like your idea, but the left-handedness makes it look like a smiley face with a unibrow to me. curse learned reading patterns

  23. You can just use a :D for the smiley, instead of the parenthesis. Or, (and I know most geeks hate this, but it works) enable autosmiley. The little burst of yellow is annoying, but at least it’s upright.

  24. “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.”

    But he wouldn’t want to, because the time-travelling DeLorean doesn’t move in space during a time jump. He’d wind up smashing into the ground at 89+ MPH sometime in the future.

  25. Nice things you try and think about when you go sled. xD
    I like this.
    Interesting that you use meters instead of inch/feet/miles.

    Some additional infos about the velocity:
    The recommended speed (depends on traffic. in a jam it’s better to drive slower) on highways in Germany is 130 kph (~81,25 mph; 36,1 m/s). The highest allowed speed ever is 250 kph (156,25 mph; 69,4 m/s) but you can’t mostly never reach this. But with a good car and a free road 150-180 kph (93,75-112,5 mph; 41,6-50 m/s) is no problem.

  26. Great post – I found it while searching for sledding hills in Cambridge. Took my 3 year old daughter out for her 1st time sledding 2 weeks ago. By the way I love how the hill ends in a pond, adds that nice touch of excitement.

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  28. they actually use gravity to activate a flux capacitor in episode 11 of season 1 of the canonnically disputed Back to the Future Animated Series (circa 1991)

  29. Given that it’s never established as 88mph relative to *what*, I have to wonder if a wind tunnel could activate a timejump. And then whether it would be possible to build a teeny tiny jet turbine to play over the time circuits as needed…

  30. I realize I’m late to the party, but the chemist in me can’t resist pointing out…
    @Andrew Humid air is not more dense than dry air; it’s actually less dense. The number of gas molecules/volume of gas is fixed. This means in humid air, you have exchanged some nitrogen, oxygen, or carbon dioxide molecules for gaseous water molecules, which actually weigh less. (Add up the molecular masses and check if you don’t believe me.) Less weight per unit of volume = less dense.

  31. No slope though, so it makes sense that the sled would only decelerate at a slower pace. But for the sixteenth of a second it would take to fall through the iced section and hit the water below, the sled would have a momentary boost in total velocity (again assuming that the ice is perfectly horizontal.) Plus, in a really shallow pond (That is, the depth of the water is less than the height of the sled) would the sled go faster, even in liquid water? Hydroplaning?

  32. In the diagram, the equation used (for the sledding problem) is rounded. the actual equation derived from ?F=MAy is V=?(2gh) where “g” is ~9.80665 m/s² and “2g” is actually 0.3867 m/s² less than 20 m/s². As a result, the answer from V=?(20h) is larger than the true solution.

    (Randall, I think that you already know this)

  33. Good info thanks for sharing with us.Nice information, valuable and excellent, as share good stuff with good ideas and concepts, lots of great information and inspiration, both of which we all need, thanks for all the enthusiasm to offer such helpful information here…

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