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.

169 thoughts on “Sledding

  1. “Since menopause (I’m 58), my once-shiny hair has become a frizz-ball. My hairdresser recommended the Nioxin System 4 shampoo for my fuzzy, thinning, color-treated hair. I’ve only used it 3 times, so I can’t attest to its effectiveness for thickening the hair, but from the first use, it made my hair soft, silky, shiny. Even before I use conditioner, my wet hair feels completely different – like I’ve already conditioned it. And when dry, it approaches my former beautiful locks. On, it says Nioxin shampoo is somewhat toxic (a 7 out of 10), with ingredients linked to cancer, etc. I only wash my hair once a week, so I’ll probably keep using it, but I thought I should mention that drawback.”

  2. “Tackling Thinning Hair on All Fronts”
    Nioxin Reviews

    “My hair had gotten progressively thinner with age and I was on the verge of buying a hair piece to deal with the baldness on the top of my head. I bought this at the recommendation of my hair stylist who said she’d heard only good things about Nioxin products. For over 4 months now, I’ve been using Nioxin System 4 Cleanser and Conditioner, along with a generic form of topical Rogaine; Hair, Skin, and Nails formula vitamins; and Oil of Evening Primrose. For the first time in over 12 years, I have a full head of hair. No more peek-a-boo scalp. I am so grateful that my condition has responded to this regimen and I have no idea whether it’s the combination of all the products or specifically just one or two items. Personally, I don’t care. I’ve made this course of treatment part of my everyday care because the result is absolutely worth it!”

  3. Pingback: Blizzard Nemo « A Still More Glorious Dawn Awaits

  4. This is a nice content with lots of information. It’s a very excellent idea for raising money for charity. I think honest review is more important for authors. I like your whole discussion. Keep it up.
    Keep blogging.ur site is good.and finelly ur site going to in high position

  5. This is a nice content with lots of information. It’s a very excellent idea for raising money for charity. I think honest review is more important for authors. I like your whole discussion. Keep it up.
    Keep blogging.ur site is good.and finelly ur site going to in high position

  6. One of the modern methods to increase community awareness of the Internet. Nowadays, the Internet has been a very good position in this field achieved.
    I’m up for being in a particular field, your information will go to the Internet. Because this method is very simple and reliable. Glad that the visitor field increases conversancy There are sites that.
    Thanks for a very nice content site

  7. A good informative post that you have shared and appreciate your work for sharing the information. Got some interesting information and would like to give it a try. Appreciate your work and keep sharing your information.

Leave a Reply

Your email address will not be published. Required fields are marked *

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>