[rerdavies]

Umm.

But the water droplets/sand particles do NOT form a straight line if you accelerate the jug. You have to decelerate the jug with exactly the right timing to form a straight line, not accelerate it. Nor do the beads in the Caltech video form a straight line, despite their dishonest attempt to arbitrarily impose a straight line on beads that are clearly not in a straight line.

In the second diagram, which is alleged to be an attempt to plot acceleration due to gravity to the position of an accelerating water jug, the values [0,1,2,4,8] are marked on the x-axis, and [0,-1,-2,-4,-8] on the y-axis. Da Vinci then plots lines from [0,-8] to the points (0,0),(1,0),(2,0),(4,0),(8,0). Doing so doesn't really establish any sort of relationship between accelerating jugs and acceleration due to gravity, even allowing for an incorrect equation of motion for an accelerating object.

Da Vinci spent most of his early career trying to sell military technology to potential patrons (largely if not completely unsuccessfully). One of the pieces of technologies he was trying to sell was a process for calculating improved artillery range tables (tables of elevation vs. range). He didn't manage to sell that either.

The second diagram is more easily interpreted as a doodle that makes an unsuccessful attempt to scry a relationship between elevation and range for artillery pieces.

[SamBam]

> You have to decelerate the jug with exactly the right timing to form a straight line, not accelerate it.

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Edit: The quote above, and my original comment below, are wrong. The article is correct. Accelerating the jug at g will create a straight line.

Corrected acceleration model: https://scratch.mit.edu/projects/805927382/

Thank you hannasanarion for the correction.

Indeed, this can be thought of intuitively (though it is slightly counter-intuitive): as the article says, if the jug is moving at a constant speed, the drops will actually make a vertical line under the jug. This is because each drop will be move with the same horizontal speed as the jug.

A decelerating jug, as proposed above, would actually create a backwards line or curve. The drops at the bottom works actually be ahead of the jug.

An accelerating jug is the only way you can get the bottom drops behind the jug.

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My previous comment:

Indeed.

I was pretty sure this was correct, but wanted to model it to confirm. On a Chromebook right now, so Scratch was the easiest way to model and share:

Accelerating jug (cat): https://scratch.mit.edu/projects/805898350/

Decelerating jug: https://scratch.mit.edu/projects/805899100/

[hannasanarion]

In your program, each of the balls starts with zero horizontal velocity. This is not analogous to real life, in which the droplets from the jug will have the same horizontal velocity as the jug at the moment they detached. They shouldn't fall straight down, they should follow a ballistic arc. I don't have the skills to set up a simulation of this right now, but I wouldn't be too surprised if the total of those ballistic arcs forms a straight line.

[SamBam]

You're totally right. Adding in the ball's horizontal speed makes the article's statement correct: you do have to accelerate the jug by g in order to get a straight line.

Corrected model: https://scratch.mit.edu/projects/805927382/

[coda_]

That's what I thought as well... But I think what the diagram isn't showing is that the drops are actually moving horizontally to the left at the speed that the jug was moving at the time the drop left the jug. This means each one is moving to the left slightly faster than the one before it... And as they fall this creates an increasing horizontal distance between them... The line is kept at a constant angle due to the fact that the earlier drops are falling downward slightly faster than the later drops. The horizontal and vertical difference between the adjacent drops maintains a 1 to 1 ratio.

[s1artibartfast]

>But the water droplets/sand particles do NOT form a straight line if you accelerate the jug. You have to decelerate the jug with exactly the right timing to form a straight line, not accelerate it.

You can make a right triangle via either acceleration or deceleration at the same rate as gravity. However, you can only make an Isosceles Right Triangle with acceleration

Vertical position is a function of t^2. If you have constant horizontal velocity, you get a concave curve like this: https://www.wolframalpha.com/input?i=Y%3D-1x%5E2

Accelerating the horizontal velocity leads to a triangle like this: https://www.wolframalpha.com/input?i=Y%3D-x

[SamBam]

Actually a constant horizontal velocity will make a straight vertical line of droplets under the jug. This is because the drops will have the same horizontal velocity as the jug.

You can see that here: https://demonstrations.wolfram.com/TrajectoryOfABomb/

At any time, the first bomb is always directly under the plane. This means all subsequent bombs will also be under the plane.

[s1artibartfast]

That's true. In the zero acceleration case you get a vertical straight line. If you have acceleration or deceleration equal to G, you get a straight angled line. If you have any acceleration or deceleration that doesn't match G, you get a curve