How would you scale this up or down

Factual Questions
1

Suppose I have unlmited space and resources to build the object I will describe.

This would be a gravity track to roll a ball bearing down and then see how far they launched. The track would be a radius of a circle the high end would be at 90 degrees and the launch end at 45 degrees. I would release the ball bearing on the high end and measure how far it traveled from its starting point. The starting size of the radius would be just big enough for the bearing to fall off the end of the launch side of the track and go straight down.

My challenge would be to find a method of scaling up that would give me some kind of predictable if not linear increase in distance as the radius size increased

Would I scale up the size of the ball? Would increases in size simply be a fixed percent? Would I look for a correlation between increase in distance and increase in size that matched each other? There seems to be an almost endless amount of possible combinations that might show a pattern but I have a feeling it is allready all fiugured out somewhere.
2

I really have no answers to add, but I am interested in this question, also.

I have this feeling that there is going to be a fixed answer to some parts of this question: such as, the weight of the ball doesn’t matter, the size of the track doesn’t matter, etc.

It reminds me of a question I posted a couple of years ago about the pressure on sandbags on top of a dam. I thought the volume of water pressing against the sandbags mattered, but it turned out the ONLY important factor was depth. The volume behind it didn’t matter at all.

J.

p.s. now that I think of it, I think the weight of the ball doesn’t matter (ignoring wind resistance). Any projectile leaving the end of a ramp at a particular speed will describe the same parabola. Gravity is constant. Consider dropping 2 weights from a tower: they fall at the same rate, ignoring wind resistance.

Do the projectiles leave the ramp at the same speed? Again, the only acceleration is due to gravity, which is constant, so, ignoring wind resistance, I THINK they will accelerate the same on the ramp and leave at the same speed.

Wind resistance is going to be significant, though, on some aspects. A dense, heavy ball will be affected less by wind resistance than a less dense ball. So the apparent results will favor the dense, heavy ball. If these experiments are done in a vacuum, the size or density of the ball shouldn’t matter.

So it seems like the only factor that will make a difference is the size of the ramp, which gives more time for gravity to accelerate the ball down the ramp.

(ooh, this is fun!) :slight_smile: Anyone else want to check my work?

3

I agree with you that the ball size should matter, a few years ago I had to scale up a bow and arrow, that was interesting but a lot more straight forward.

4

I get the final velocity, v, to be

v = (2)^(1/4) x (gxR)^(1/2) (where g is the gravitational constant and R is the radius of the circle.
The velocity will always be 45 degrees to the plumb line at the time of the ball leaving the segment.

5

If you neglect friction, the size of the ball does not matter as long as (r/R) - the radius of the ball versus that of the circle is small.

If r/R is large then the equation will be

v = ((2)^(1/4) x [g x {R-(r/sqrt(2))}]^(1/2)

6 
So there really would be no scaling up or down, the only thing that would matter would be how far the ball dropped before it was launched.
7

The loss in potential energy is the gain in kinetic energy. So yes - thats the basic idea.

Now there is some scaling you can do with making the ball hollow - and recovering some of the kinetic energy as the spinning of the ball itself (This spinning energy is not shown in the above equations).

A hollow ball made of a high density material (lead for example) can be used.

8

Making the ball hollow would actually decrease the maximum distance, as you’d get a larger proportion of the energy in useless rotational motion rather than the translational motion you want. For the ideal case, you’d want a projectile that slides instead of rolling (make the track out of Teflon), or that failing, something with a small, very dense core surrounded by a larger lightweight rolling surface. In such an ideal case, the range (defined as the point where it drops below the height of the launch point) will be twice the net drop height.

And as a nitpick, you repeatedly used the word “radius” in the OP when you meant “arc”.

9

In one of my high school science classes, we had a competition for the fastest “car” rolling down a track. Most teams seemed to focus on lubrication and smoothly rolling wheels. I put two CDs on a heavy bolt (spaced apart so it could roll like a bobbin). The low friction was useful but more important was the large radius compared to the moment of inertia. We won by a wide margin.

10

So the speed is roughly related to the
Have you tried to shoot dust ? Doesn’t go very far, aerodynamics stops it real quick.

Why ? well mass is related to r^3. but the surface area is r^2. So when its really small, it has a very small mass compared to surface area… Go the other way, its getting more mass compared to surface area.

Well the speed its traveling is the same ,because it dropped down the same height.

But then it gets slowed by aerodynamics, so larger is better for that.

The CD’s held by bolt has quite small aerodynamic profile too. The slippery CD’s also prevented the wheels gripping and smearing into the surface… too much grip is a problem in that the grip sucks up energy and gets hot …wasting the energy. The lesser number of greasy bearings meant less grease drag too. The grease is meant to get hot but on such a small run, the energy of heating it is significant.