After I read this mnemonic it just didn’t feel right, would the arc be steeper? Galileo proved that all objects fall at the same rate in a vacuum. It got me thinking whether an object’s weight would really affect the arc’s curvature when launched from some theoretical bow. Given the same launch angle and no air resistance, I would say no it doesn’t matter how heavy an object is, all objects would produce the same arc just with varying sizes. Regardless of the effectiveness of the mnemonic, I thought it would give rise to some interesting discussion.

Also wanikani shaming a melon for being too heavy to fly straight, implying that any object that comes out of a launcher flies straight.

well I have never formally studied physics, but from what I understand the melon and arrow would fall at the same rate but the arrow would travel further before hitting the ground, so the melon’s arc would slope down much steeper than the arrow in terms of horizontal distance/vertical distance.

Right?

*editing to add that I suppose in the end the melon’s distance covered all has to do with how much force the melon is shot with, but if we are imagining a standard bow, it’s not gonna get very far.

Then yes! You are entirely correct! The melon will follow a parabolic curve, and much like circles they are all the same. The circle only has one property, the radius, and the parabola only has the focus length. All parabolas are essentially the same

If you account for air resistance, won’t the path of a melon have lower average curvature than lighter objects of the same size and shape if launched with the same velocity or even momentum? I.E., melons fly straighter?

Edit: Should have said this only holds after rescaling the paths to try to be as similar as possible. Obviously, if you launch two objects with the same momentum and don’t rescale their paths, the heavy object will have a steeper angle of impact.

Isn’t this the big differentiator though, generally, people don’t throw their melons into perfect vacuums. If I were to do as the mnemonic says right now and shoot a melon out of a bow, I can be sure that there would be air resistance acting on it. In which case mass does have an effect:

Gravity is still easy: F_g = m \cdot g
Air resistance: F_a = \frac{1}{2}A \cdot \rho \cdot C_D \cdot v^2

The total force will then be F_g - F_a, which by netwon’s second law will result in:

a = \frac{F_g - F_a}{m}

Now if F_a = 0, we can divide m out of the formula resulting in an acceleration of g, which would be the vacuum case, in the none vacuum case, you can’t just discard the mass in its entirety. We can also write acceleration as a = g - \frac{F_a}{m}, in which case the heavier the object the less of an effect air resistance will have. Although increasing the acceleration ends up increasing velocity which increases air resistance quadratically, which would eventually result in the object reaching terminal velocity. Although if your mass nears infinity it would probably cancel out (assuming the planet doesn’t start falling towards you), so just use a really heavy melon.

Given the same angle and the same velocity, all objects will follow the same arc. But a melon launched from the same bow as you use with your arrows is just gonna land at your feet, because tension on the string is going to impart a lower acceleration on the heavier melon.

But the air resistance isn’t always in the same direction as the weight force so you can’t just add those two together.
Besides, if the angle of launch was the same and the objects didn’t reach terminal velocity wouldn’t the arcs be the same but smaller (since all air resistance is doing is taking some of the objects’ energy)? I was thinking more of dismissing wind with that assumption than the overall distance reduction.

So I’ve drawn a picture which should show what I’m trying to describe more clearly. Weight wouldn’t matter, the overall shape shape is still the same, or so I think so far.

Well yeah, neglecting air resistance, the path followed is always going to be a parabola.

When you’re comparing an arrow flying over the horizon to a melon dropping at your feet, anyone you ask is definitely going to describe the latter flight path as “more steep”.

Ah yes silly me, how could I forget such a crucial assumption!

A healthy dose of skepticism my good sir.

Melons fly straighter? They sure do when I load them into my cannon. Nice to see someone having a completely different viewpoint. I believe if they’re launched at the same angle it wouldn’t matter.

Sure makes calculations a hell of a lot easier.

I’m not dismissing that, but wouldn’t the velocity only change the size of the arc? The fundamental shape remains the same as if just being shrunk.