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Sledding: Newsletter Challenge (02/26/08)

Posted February 24, 2008 5:01 PM

The question as it appears in the 02/26 edition of Specs & Techs from GlobalSpec:

You give a sled a quick push up a snowy slope that makes an angle θ with the horizontal. In time t, the sled goes up and comes back down. If the coefficient of friction between the sled and the slope (snow) is µ, determine the time it takes the sled to reach the top of its trajectory.

(Update: March 4, 9:00 AM EST) And the Answer is...

The following diagram shows the force balance when the sled is going down. A similar diagram can be drawn for the situation when the sled goes up. Notice that the x and y axes are drawn along that slope and are vertical to the slope, respectively.

The weight of the sled is given by

and the force of friction (along the slope going up) is given by

where is the friction coefficient and is the normal force

The weight components along the axes are given by

Now apply Newton's second law to the y-direction:

Notice that because there is no motion in the y-direction, . Therefore,

(1)

Similarly, then by applying Newton's second law to the x-direction, we get

Canceling the mass that appears in each term, we get an equation for the acceleration along the x-direction (down the slope):

For clarity, let's rename this acceleration as . Thus,

(2)

For the case when the sled goes up, we can get a similar equation for the acceleration (I'll leave the derivation as an exercise for my reader!):

(3)

We know that for a constant acceleration the speed is expressed using the kinematic equation of motion given by

where and are the initial speed and initial position, respectively. We see that the acceleration is directly proportional to the square of the velocity at any point in the sled trajectory. If and are the initial velocity when the sled is moving up and the final velocity when moving down, respectively (notice that in both cases they represent the same position in the trajectory), then we can write:

(4)

The speed can be expressed as where is the acceleration and is the time. Let's define as the time the sled reaches the top, and as the time it takes to move down. By substituting the velocities in (4) with their respective equations we get

(5)

Or,

For the purpose of simplification, let's define

(6)

Thus,

(7)

We know that

Using Eq. (7), we get,

Finally, the time it takes for the sled to reach the peak is given by

(8)

Note 1: It is clear that we must have

Note 2: It also should be clear that if there is no friction (), then

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#98
In reply to #97
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Re: Sledding: Newsletter Challenge (02/26/08)

02/28/2008 5:00 PM

The challenge specifies a constant coefficient of friction, so that is what we use. I think we appreciate that that is an extreme simplification. By the way, this view of dynamic friction corresponds to the squarest hysteresis loop you can imagine.

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#99

Re: Sledding: Newsletter Challenge (02/26/08)

02/29/2008 12:05 PM

Easy question!

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#100

Re: Sledding: Newsletter Challenge (02/26/08)

02/29/2008 5:54 PM

Well done Tkot and BobD, I agree with your answers too.

The above diagram shows a plot of the ratio "time up"/"total time" Vs. angle, as given by both Tkot and BobD, with an arbitrary value of μ=0.2.

As you can see the value of μ dictates a critical angle (About 12 degrees in the case of μ=0.2, below which the sled will never return ( I.E it gets stuck at the top of it's trajectory). This is because, below this angle, mgμcosθ becomes greater than mgsinθ (The friction force then exceeds the gravitational force) .

Also note that in the extreme case of a hill which is at 90 degrees (μcosθ now equals zero regardless of the value of μ (I.E. no friction and therefore time up = total time/2.

This 90 degree case of course is akin to throwing an object vertically upwards with no losses to friction or drag (I.E time up will equal time down).

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#101

Re: Sledding: Newsletter Challenge (02/26/08)

02/29/2008 6:48 PM

More trivia charts for anyone interested :)

The above chart shows how the critical angle, in degrees, varies as μ goes between 0 and 1. Again, worthy of note here is that when μ = 1 , the critical angle occurs when sinθ=cosθ, I.E at an angle of 45 degrees

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#102

Re: Sledding: Newsletter Challenge (02/26/08)

02/29/2008 6:59 PM

More trivia charts for anyone interested :)

The above chart shows how the critical angle, in degrees, varies as μ goes between 0 and 1. Again, worthy of note here is that when μ = 1 , the critical angle occurs when sinθ=cosθ, I.E at an angle of 45 degrees

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#103

Re: Sledding: Newsletter Challenge (02/26/08)

02/29/2008 7:20 PM

More trivia charts for anyone interested :)

The above chart shows how the critical angle, in degrees, varies as μ goes between 0 and 1. Again, worthy of note here is that when μ = 1 , the critical angle occurs when sinθ=cosθ, I.E at an angle of 45 degrees

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#104

Re: Sledding: Newsletter Challenge (02/26/08)

02/29/2008 7:26 PM

My humble apologies for submitting this last chart multiple times.

I forgot to click <next> after submitting the entry and kept thinking the post had failed

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#105

Re: Sledding: Newsletter Challenge (02/26/08)

02/29/2008 9:42 PM

Fig 1 and Fig 2 above show why the forces are different when the sled travels up and down. On the way up gravity and friction forces are adding to slow down the sled quickly. On the way down the force is reduced because the force of friction changes direction and must now be subtracted from the force of gravity to get the overall force in the downward direction. This reduced force results in a smaller value for acceleration down than the decceleration value up and hence the rate of change of velocity down is smaller than that going up. Graphically we can see this if we look at how the sled's velocity is changing with time:-

The velocity graph on the way up forms a triangle as the velocity goes from an arbitrary initial value at time zero to zero velocity at that time when the sled reaches the top of it's trajectory. The second triangle is formed as the sled slides back down but you will note that the rate of change of velocity is much smaller now. The slope of these two triangles is actually the decceleration and acceleration values respectively:-

I.E. slope of first triangle = g(sinθ + μcosθ)

and slope of second triangle = g(sinθ - μcosθ)

Now the one thing we know for sure is that the distance up = distance down.

The distance up is simply the area of the first triangle and the distance down is the area of the second triangle. The triangles will get uniformly bigger if the initial push gets bigger but they will always have the same shape for any given value of μ and θ.

Now hopefully, anyone who is still in doubt, will begin to see that the ratio of time up to the overall time is not dependent on the initial velocity or strength of the push, it is only dependent on any given μ and θ.

I do hope this helps clarify the results rather than further confuse!!!

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#106
In reply to #105

Re: Sledding: Newsletter Challenge (02/26/08)

03/01/2008 3:10 AM

The graphics are good, the only bit missing is the bit clarified by BobD. Namely that distance covered up and down is 1/2.a.t2 Dropping the illustrated accelerations into it and doing a bit of re-arranging gives the resultant time ratios. I give you a vote for taking the time to present the visuals.

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#107
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Re: Sledding: Newsletter Challenge (02/26/08)

03/01/2008 12:52 PM

The visuals in this and your previous posts are great!

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#108
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Re: Sledding: Newsletter Challenge (02/26/08)

03/02/2008 1:23 AM

Hey, Ken. Fancy an upgrade !

340 mph - not surpised it's got those wheels at the back, and being so close to the ground must make it feel like 1000. The 'pucker' effect would make seatbelts pointless.

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Re: Sledding: Newsletter Challenge (02/26/08)

03/02/2008 11:20 AM

My trike actually started off on two wheels, with foot-controlled outriggers. In fact, the cutout you can see near the nose was for front wheel clearance. (I decided not to close up the cutouts on this proof of concept prototype -- for fun, mainly -- it makes people wonder: what are those for? Aero in this one is not too important, with the "real thing" being larger.)

Two wheels can hardly be beat for aero efficiency. The rear wheel is more-or-less in the wake of the front wheel, whereas on a trike, each wheel creates its own wake. But with computer-controlled or manual outriggers, the transition from very low speed to ordinary driving speeds is still a bit tricky, causing a bit of weaving, if you're not focused on what you are doing. So, given that my vehicle will be driven by people early in the morning before the coffee has kicked in, I decided to go with three wheels. (But still, at some point I'd like to build another enclosed two wheeler -- but not for commercialization.)

My next prototype will look like this:

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#110
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Re: Sledding: Newsletter Challenge (02/26/08)

03/02/2008 12:03 PM

I've been puzzled over your pic for a while. It's ultra-cool. The picture I posted was something I saw in a magazine recently. The Acabio GTBO. I don't know much about it, but they seem to be into concept behicles, and despite the jaw-dropping performance of some, they have a seemingly serious agenda about fuel efficient transport. It was a very tenouous link to mention it in relation to sleds, but I'd be interested to hear if the topic of vehicles like yours has a home on CR4. Low-profile, economy efficient transport is a very cool topic. In my wonderings I've not seen the appropriate thread. Any pointers ?

Cheers, Kris

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Re: Sledding: Newsletter Challenge (02/26/08)

03/03/2008 4:41 PM

Hi Kris:

There have been a couple threads on the Aptera (which is a bit heavier than mine, a lot more expensive, but otherwise similar) but not many others. My guess is that by the time the Automotive X Prize gets into full swing, there will be about 30-40 teams working on very small vehicles (typically three wheel) and a similar number working on larger, four wheeled vehicles.

The way that fuel efficiency is measured on these vehicles can be really bizarre.

The actual energy consumed by a vehicle can be calculated within pretty close limits if you know weight, aero drag coefficient, rolling drag coefficient, etc. So a REAL 100 mpg vehicle (one that you can fill with one gallon of gas and go 100 miles, without any external energy source) ends up being small and light -- their's no magic. But once there are two or more energy sources, marketers find it convenient to ignore as many as they can. (It reminds me of the scene from The Wizard of Oz "Ignore that man behind the curtain." )

This manner of thinking, in which only one source of energy is accounted for, is promoted by a previous CIA director turned ethanol advocate, Jim Woolsey. In his logic, you first make a 50 MPG hybrid mid-sized car. So far so good -- just a slight improvement on a Prius. Then you make it plug-in by adding some more batteries and a charger. Then you rig the test length to use mainly batteries, and almost no gas, and calculate a believable figure, like 90 mpg. Finally, you include only the 15% of E85 that is actually gasoline in your fuel efficiency rating: 600.3 mpg! He introduced this concept in a wall street journal op-ed piece, where he called it a 500 mpg car -- apparently so he could say something like "the 500 mpg is a conservative estimate"

I'll plan to gather up some links to some of the new small vehicles coming out and start a thread soon.

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Re: Sledding: Newsletter Challenge (02/26/08)

03/04/2008 2:47 AM

Hi ken,

Thanks for the info and links, I've bookmarked them to read-up on. There's certainly a lot of jiggery-pokery involved in published fuel economy, it's another one for the 'lies, damned lies....' folder. No doubt the manufacturers will get even more inventive as green issues become more important. It would be great to read any new thread you can put together on the topic.

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#116

Re: Sledding: Newsletter Challenge (02/26/08)

03/06/2008 11:13 PM

A mathematician named Fred
Loved sliding downhill on a sled
When he learned about μ
He knew just what to do
So now he slides uphill instead.

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#117

Re: Sledding: Newsletter Challenge (02/26/08)

03/07/2008 12:10 PM

Well, if we're going to be poetic...

On a forum called CR4.
Were challenge questions galore.
Some saw no use,
But others, an excuse
To get on Google and explore!

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