Gravity Vehicle C

[chalker7]

I think there is some confusion here about what Brachistochrone means exactly. It just means the curve that minimizes the speed for an object rolling of any shape. You can still have a brachistochrone curve that will maximize potential energy (that is ~1m tall).

Why would you want to minimize speed?

I apologize, I mistyped. It minimizes the TIME to travel down the ramp and maximizes the speed of the rolling object by increasing its acceleration during the early stages of rolling.

[Dabbler]

ohhh that makes sense! Thanks!

[chalker]

I apologize, I mistyped. It minimizes the TIME to travel down the ramp and maximizes the speed of the rolling object by increasing its acceleration during the early stages of rolling.

That's your one mistake you're allowed this year, next time you're kicked out of the family;)


On a serious note though, according to the wikipedia page on Brachistochrone curves (http://en.wikipedia.org/wiki/Brachistochrone_curve), they are applicable to POINT-LIKE bodies sliding down the curve. Obviously the car will have at least 2 axles that are a certain distance apart, and thus not be point-like. As a result, I suspect a different type of curve will be more ideal. Anyone have any thoughts on this?

[buzzbuzz]

A different curve may be more ideal, but I think we should all put more time into perfecting the time prediction and accuracy of the vehicle rather than focusing on changing the ramp to save maybe a few tenths of a second.

[Balsa Man]

An interesting question.
Still working thru the physics, but my initial take is as follows:

In an idealized model, the shape of the ramp shouldn't affect the final speed.

mass * (gravitational acceleration) * height = potential energy at the top = kinetic energy at the bottom = 1/2 * mas * (velocity)^2

Mass cancels out, so the final velocity would be sqrt(height * (gravitational acceleration))

That suggests the ramp shape should be designed to both:
(a) maximize the initial height of the center of mass of the vehicle (maximizing the initial potential energy) and
(b) minimize the energy lost due to friction and bouncing

(a) suggests that it should start off fairly flat at the top (with the rear wheels sitting right at the top edge of the ramp, 1m above the floor, the closer to vertical, the lower the center of mass would be), and I think (b) means it should be reasonably flat at the bottom as well (with a curved transition to flat). So, my first impression is that it should look something like:
_
_/


There is another place that the potential energy will be going: the rotating wheels. Still working thru the implications of this. If you have big heavy wheels, then a significant part of the initial potential energy will be transformed into rotational energy in the wheels, which would imply less kinetic energy and velocity, and suggest, Ideally, the wheels should be massless, but there is obviously some trade off here with the stability of the vehicle. It would also the rotational energy (momentum) of the wheels would end up (minus rolling and axle friction) re-converting to kinetic (horizontal) energy as it rolls along the floor.

Thoughts?

[chalker7]

An interesting question.
Still working thru the physics, but my initial take is as follows:

In an idealized model, the shape of the ramp shouldn't affect the final speed.

mass * (gravitational acceleration) * height = potential energy at the top = kinetic energy at the bottom = 1/2 * mas * (velocity)^2

Mass cancels out, so the final velocity would be sqrt(height * (gravitational acceleration))

That suggests the ramp shape should be designed to both:
(a) maximize the initial height of the center of mass of the vehicle (maximizing the initial potential energy) and
(b) minimize the energy lost due to friction and bouncing

(a) suggests that it should start off fairly flat at the top (with the rear wheels sitting right at the top edge of the ramp, 1m above the floor, the closer to vertical, the lower the center of mass would be), and I think (b) means it should be reasonably flat at the bottom as well (with a curved transition to flat). So, my first impression is that it should look something like:
_
_/


There is another place that the potential energy will be going: the rotating wheels. Still working thru the implications of this. If you have big heavy wheels, then a significant part of the initial potential energy will be transformed into rotational energy in the wheels, which would imply less kinetic energy and velocity, and suggest, Ideally, the wheels should be massless, but there is obviously some trade off here with the stability of the vehicle. It would also the rotational energy (momentum) of the wheels would end up (minus rolling and axle friction) re-converting to kinetic (horizontal) energy as it rolls along the floor.

Thoughts?

While that is theoretically correct, you are skipping the part where the energy actually gets transfered from vertical acceleration/velocity to horizontal. Specifically, you're neglecting the fact that downward force component (that which provides the acceleration) changes depending on the angle of the ramp. Draw a force diagram with both a shallow ramp and a steep ramp to see how the different forces play out and affect final speed.

[bearasauras]

Chalker, let me know if Chalker7 is kicked out of the family. I'd like to apply for that position.

Regarding maximizing the potential energy at the start and following the Brachistochrone. I think the confusion comes from people looking at the picture of the curve and not understanding where it comes from. Here's a picture of the curve for going 100 units down and 75 units across. Like Chalker said earlier, you can have a brachistochrone and still maximize your potential energy.


Image generated by a Mathematica® file found online.

As for the vehicle not being a point mass. Yup, that's a valid concern, but it really depends on the design of your vehicle. Of course, if your vehicle's short, the center of mass of the vehicle will follow the curve of the ramp pretty well. But now, imagine your vehicle to be as along as the ramp (connecting the top of the ramp to the bottom of the ramp), and you put all your mass at the back (higher) end of the vehicle. Assuming that the back wheels are pretty close to the back of the vehicle and that they aren't ginormous, the center of mass of your vehicle will still follow the curve of the ramp pretty closely. On the other hand, if you have a really long vehicle that once again connects the top of the ramp to the bottom of the ramp, depending on the ramp's curvature, the center of mass of your vehicle will most likely not follow too closely to the ramp's curvature.


The following will probably be not applicable for most participants, but if you're interested or just want to know about it, read on!
The Brachistochrone curve can be derived using a couple of different methods. If you've learned calculus and is interested in looking at some calculus of variations (it's not the easiest thing on Earth), you can actually derive the Brachistochrone curve for any given point on your own. And while doing that, you can add the constraint given by where the mass sits with respect to the two wheels. This calculation will give your the Brachistochrone curve for your system (ignoring friction). Of course, you can add in friction too, but that's beyond me since it would be pretty hard to model friction. That's where testing and modifying the ramp would come in really handy.

[Balsa Man]

In an idealized model, the shape of the ramp shouldn't affect the final speed.

mass * (gravitational acceleration) * height = potential energy at the top = kinetic energy at the bottom = 1/2 * mas * (velocity)^2

Mass cancels out, so the final velocity would be sqrt(height * (gravitational acceleration))

?

While that is theoretically correct, you are skipping the part where the energy actually gets transfered from vertical acceleration/velocity to horizontal. Specifically, you're neglecting the fact that downward force component (that which provides the acceleration) changes depending on the angle of the ramp. Draw a force diagram with both a shallow ramp and a steep ramp to see how the different forces play out and affect final speed.

First. let me say, when I first thought about this, my take was consistent with what you're saying.
Perceptions can some times get in the way of physics, though, so I stand by the physics I noted above.
First, re: shallow vs steep ramps; given the dimension constraints of the rules (1.0vert/0.75horiz), there is only one (flat) ramp that will fit; let's say top at A, and bottom at B; ramp is line from A to B.
You can, of course, also run a brachistochrone curve between A & B. Comparing the two, with an...."object" moving down under gravity, ignoring friction (and the instantaneous transition from sloped to flat on the flat ramp) - A brachistochrone curve does provided the shortest time path for an object.

What seems to be at question here, and it's the hard part to get one's head around, is whether that means an object coming off the brachistochrone curve ramp will have a higher horizontal velocity than the object off the flat ramp. Psychology/human perception bumping into physics. It would seem so, but, as I now understand it, that is not how it works

Here's an interesting paper- see page 2 onto 3 particularly.
http://uweb.cas.usf.edu/~drohrer/pdfs/Rohrer2003M&C.pdf

If someone can shoot a hole this, and the basic physics (that I noted, and that are reflected in this paper), I'd really like to see it. I'm thinking about putting a couple simple ramps together this weekend for marble testing to confirm this.....non-intuitive conclusion.
There are a lot of other pieces to the puzzle of optimizing a gravity vehicle; getting the right trade-offs of maximum velocity, and minimum lateral and linear error, and predictability/repeatability- all together. Fun stuff!

[illusionist]

Just wanted to say to those who said that this was a simple event, read the last 10 or so posts =P
And what bear said is true. If you do have a really long vehicle, but concentrate most of the mass at the highest point (the back of the vehicle) then it will be similar enough to a single point following the ramp shape. However, I believe that a shorter vehicle will work best with a curved ramp shape.

[sj]

Actually i believe, and correct me if im wrong, that a vehicle which is as long as the ramp, that is the front wheels at the front edge and back wheels at the top edge, would be the best imitation of a point like body on a curved ramp. This is because once the car starts rolling the front wheels are off the ramp and only the back wheels are rolling down the ramp. Therefore if the majority of the mass was concentrated about the center of rotation ( the axle) of the back wheels the car would in fact behave closely like a point like body.

Also must the car also be inside the 1 meter x 75 cm box or are the back wheels allowed to sit on a flat surface level with the 100 cm mark at the initial position?