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.
The obvious follow-up question is then where would Marty McFly get the necessary 1.21 GWs (or Jiggawatts if you prefer the film script spelling or if you play too much Banjo Kazooie) of electricity to power the flux capacitor during his 40 m/s jump?
I thought 88 miles per hour was just a speed to trigger the flux capacitor, which would then propel the Delorian to light speed (which is where the flame treadmarks come from). So if Marty jumped, and was sent to light speed, he’d have trouble slowing down on reentry, no matter what kind of trampoline he had waiting for him.
For me the obvious follow-up question would be “what would be needed in order to make the speed of the sled greater than the speed of the average raptor?”
…but maybe that’s just me.
I would suggest some small rocket’s attached to the back of the sled. The weight of the body on top should keep it close enough to the ground.
good thought with the gps, I would have pegged you for an arduino, an accelerometer and several hours of coding
the “innards of his DeLorean” must include the original plutonium chamber or mr fusion, although as captain jack (not sparrow) can attest to, time travel without a capsule is a killer. And the ice on the delorean after the first run has me sceptical about the viability of this plan.
wouldn’t the gps only provide the horizontal speed? for an accurate reading, you’d need to take the slope of the hill into account. maybe you can outrun those raptors after all…
I rather enjoy talk of F=ma and coefficients of friction, especially when I just had my AP Physics midterm.
On a side note, what is the coefficient of kinetic friction of your mother on snow? My friend asked me this. I’m sure he was referring to my mother, but either way.
@ Koko
The presence of a small raptor should provide sufficient motivation for the sled to accelerate proportionately and also to navigate in ways hitherto unforeseen.
@koko
Maybe this is just because I didn’t like physics, but if there is snow on the ground the raptor will either be dead or in hibernation.
“what would be needed in order to make the speed of the sled greater than the speed of the average raptor?”
I expect the obvious answer is : You’re gonna need a bigger slope.
(“Slope” here is a substitute for “hill”, for better assonance with “boat”. I don’t mean the hill needs to be more vertical.)
The sydney harbour bridge is 50m high. Now i’m gonnna be thinking about swerving when i commute lol.
@ koko;
But how fast can raptors run downhill on ice without falling? Would they fall? If they did, would the slide slower or faster than a sled? What sort of traction does raptor skin have? Would it make a good handbag? Would the bag it sell on ebay? What about shoes?
These are the essential questions one must be asking.
A small pet peeve of mine. The fastest a human can run is a lot closer to 12 m/s than 10 m/s. The 10 m/s time is the average speed over a 100 m dash for the fastest humans, which includes delay in starting and accelerating.
So… you still can’t outrun a raptor using gravity?
…
DANG it.
Mike Stanton:
> On a side note, what is the coefficient of kinetic friction of your mother on snow? My friend asked me this. I’m sure he was referring to my mother, but either way.
Should we assume a spherical mom?
“I thought 88 miles per hour was just a speed to trigger the flux capacitor, which would then propel the Delorian to light speed (which is where the flame treadmarks come from). So if Marty jumped, and was sent to light speed, he’d have trouble slowing down on reentry, no matter what kind of trampoline he had waiting for him.”
—
The car is not traveling at light speed when it appears later on, I’m assuming that the car reaches light speed to make the leap, then rapidly de-accelerates as it returns, implying that he could make the fall without reaching such a crippling speed, I do fear the effects this will have on his body though, humans are not built for such rapid acceleration.
I don’t get the GPS thingy, like Ryan said “wouldn’t the gps only provide the horizontal speed? for an accurate reading, you’d need to take the slope of the hill into account”? Or your just saying that your getting the instant max velocity at the end of the slope with the GPS?
Sorry… but I have a physics exam in about 2 weeks
I’m so glad I’m not the only one using GPSr to track sled speeds. I use it sledding because it’s hard to know just how much fun you’re having unless you’re measuring.
That is, I was glad until I read Ryan’s and Lorenzo’s doubts. But since my GPSr also knows my altitude (from having acquired lots of satellites), wouldn’t its “speed” measurements be accurate in three dimensions? (Note to self: find sledding hill whose height is taller than it is long. . . .)
a somehow related video, that shows the inverse direction: given a car with some speed and a (improvised) ramp, how high can a car go.
http://tv.derstandard.at/?catid=1&vidid=2443 (there will a short commercial clip before the real video).
Yes. It’s in german, but still…
You really lose 4 m/s from snow friction? That’s actually rather significant. =\
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So where did you get a coefficient of friction for steel on snow? It’s not a commonly used thing, after all- did you do some testing to determine it?
And I agree about the possible fault of using the GPS- if it’s a steep enough hill (say, a vertical angle of 30 degrees), your actual speed relative to the slope is going to be significantly higher than your horizontal speed. V/(cos 30) gives me about 11.5 m/s slope speed, as opposed to 10 m/s horizontal.
GPS worries, if you’re calculating average speed by taking total distance over total time (only really true for a flat slope, otherwise the average speed involves integrals) you are getting the average horizontal speed and neglecting the vertical component. effectivley you have calculated the total velocity times Cos (angle on slope), or
v*(d/s) = v*(d/sqrt(d^2+h^2)
where d is the horizontal distance and h is the vertical distance (i’m assuming d is given by the gps? and you can get h from google earth/guesswork)
a better eqation would be:
v(av) = sqrt(d^2+h^2)/t
v(max) = 2v(av)
which again is a poor approximation assuming constant accn ie frictionless, constant gradient…
then check that with your GPS reading!
So what happened to the other 5 m/s?
Using an anemometer would be more accurate imo (assuming the park was not windy). GPS cannot provide top speed, only a mean speed over a second.
I’d wager the list of exclusionary states would be somewhat longer if you consider only the buildings of 30 years ago.
That notwithstanding, the Delorean innards as illustrated would probably create enough additional drag as to have a negative impact on Marty’s terminal velocity, making it so he’d never achieve the required 88 mph even if he jumped from the Taipei 101.
completely unrelated (well, maybe not completely) but you should go again tonight. Maybe after it stops raining, but right after all the new snow is down. Good times…
I can see the building where I am right now on the list!
And McFly wouldn’t need a building. He could just get the car to the top of a tall hill and coast in neutral. I got an old chevy POS up to 110 that way easily. Of course, he would lack the Giggawatts…
Actually, North Dakota has the 2nd tallest structure in the world: the KVLY-TV mast (http://en.wikipedia.org/wiki/KVLY-TV_mast). It is 628.8 meters tall.
Biggest slide evar:
http://tinyurl.com/afjbol
EVAR. I told you it was huge!
Koko asked:
“what would be needed in order to make the speed of the sled greater than the speed of the average raptor?”
If you and your friends are being chased by a hungry velociraptor, you don’t necessarily need to outrun the velociraptor.
re: McFly – I’d always wondered why Marty had to time his approach to the bell tower in 1955. Why couldn’t he have just stayed in place and revved the engine?
– A.
Worked in his state’s tallest building,
but only on the 6th floor
Hehe it looks like your comment-rendering template has some unresolved conflicts : )
Hmmmmm…. why am I seeing each of the comments twice? o.O (FireFox 3.0.5)
To everyone who pointed this out:
> wouldn’t the gps only provide the horizontal speed? for an accurate reading, you’d need to take the slope of the hill into account.
Yes, but we reach maximum speed at the point where the hill is leveling out, and grades are always waaaaay less steep than people visually estimate them to be. Since sin(x)~=x for small x, and the grade of the hill at that point is well under 5%, this correction is quite a bit less than the inaccuracy of the GPS.
> good thought with the gps, I would have pegged you for an arduino, an accelerometer and several hours of coding
Good guess! Back at Mike’s house there is indeed an Arduino hooked up to a Wii Nunchuck. But it’s not quite ready for its super-cool project yet.
GPS systems in fact calculate an x,y,z coordinate system wherein the origin sits at Earth’s center of gravity, and the vertical origin axis runs through the intersections of the prime meredian and the equator.
Two things:
1) The statement “[] When you slide down something with no friction, your speed doesn?t depend on the path you take” is WRONG. The fastest path would be a cycloid (See Brachistochrone curve in wikipedia), but the acceleration won’t be uniform, and the max speed would be somewhere in the middle of the path (not at the end).
2) @max and @alex DeLorian had a punch in the tank and was out-of-gas; hence 88mph was an issue. But the extra power of 1.21 GW was NOT an issue, since by then DeLorian had the mod for the flux capacitor that enabled it to run on rotten eggs.
3) @aaron: Cause DeLorean had to be at the precise location at the precise time (of the stoke), with speed over 88mph.
That makes three things. but hey, what the heck..
Randall,
This is where you really want to go sledding:
http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=36+eastern+ave,+arlington,+ma&sll=42.41358,-71.173897&sspn=0.003121,0.006909&ie=UTF8&ll=42.413057,-71.174133&spn=0.003311,0.006909&t=h&z=17
It occurs to me that this comic has one of the easiest times getting itself published in Wikipedia.
One a side note…
Are you perchance related to this: ( http://velociraptorz.org/ ) sinister organization?
We have you now.
Please, for the love of God, please – stop using metric.
“One of them gives the speed of an object after it?s fallen height ?h? in Earth?s gravity”
I hate to be a stickler, but shouldn’t it be “its” instead?
Also, I’ve no idea why my apostrophes showed up as question marks…
Doc Brown – how can the max speed along a Brachistochrone curve be in the middle? In the absence of friction, no energy is transmitted out of the object during its descent, therefore its speed must be directly related to the amount of kinetic energy it gains from its loss of potential energy–thus, at any point, the speed is proportional to the square root of its vertical descent. This would imply that its speed is always at its maximum at the bottom of the hill.
The total *time* it takes to get to the bottom could depend on its path–on a cycloid curve, the descent at the beginning is faster, so it picks up speed quicker, which could get it down to the bottom faster than if it goes down a diagonal slope. However, by the time it gets to the bottom, its *speed* should be the same as if it went down a diagonal, or any other path.
Where have you been, Lord Randall my son?
Where have you been, my handsome young man?
Oh I fear I am poisoned, mother make my bed soon,
For I’m sick to the heart and I fain would lie down.
Oh that was strong poison, Lord Randall my son,
That was strong poison, my handsome young man;
Oh I’m weary and worn, mother make my bed soon,
For I’m sick to the heart and I fain would lie down.
I was keen to try this in Melbourne today but it’s already 43°C (that’s a smidgen under 110°F). Sleds don’t go so well on baking, rocky hillsides.
@ Goatee Man: I believe the “it’s” in question is a contraction of “it has,” so the apostrophe would be necessary.
Clearly, if I were running the numbers on this problem, the question I would be asking is “Why does Randall post so infrequently?”
What do you need to outrun (outsled?) the raptor? Easy, a double barrel shotgun! That will make the raptor go from +5 m/s in comparison to your speed to -15 m/s is comparison to your speed. Quite quickly too I would imagine.
Keep these posts coming, it is neat to hear from you, especially the physics stuff.
@ Patrick:
Why would we ever stop using metric? The entire system is based around 10, and is virtually standardized with the exception of Liberia, Burma and the States. So please, stop using imperial.