Physics geeks: Galileo and cannonballs

Factual Questions
1

I’m copyediting a history book with a chapter on the scientific revolution. The author is claiming that

:confused:

Did Galileo really show this?
If so, was it later disproved?
Wouldn’t the cannonball’s motion also have a component equal to the rotation of the earth, because the cannon from which it was fired was also moving with the earth at the time? And thus wouldn’t the cannonball indeed land at the spot from where it was fired?

Basically, I need to know whether I should query the author to check his facts, or is this correct as stated. I’ve done some Googling, with inconclusive results; several university sites mentioned the example of a ball that was dropped from the mast of a moving ship and did indeed land at its base, which seems to disprove my author’s claim.

Many thanks!

2

You could show that at the exact point of the cannonball it has its upward motion plus sideways (tangential) motion as a result of it previously being carried along by the rotating Earth.

After leaving the cannon it travels in roughly a parabola described by ordinary Newtonian gravity. However, you could also see it as going into in a very elongated orbit around the center of the Earth (that, given enough time, simply intersects the Earth again, at which it stops ‘orbiting’, and just hits the ground).

(neglecting atmosphere and any other frictions, and assuming perfectly vertical launch, at the equator, etc.etc.)

A cannonball just dropped from the lip of the cannon is itself in an ‘orbit’ that has tangential motion equivalent to the Earth’s rotation at that point, but of course it hits the ground because its effective orbit intersects the surface of the Earth.

Okay, so trying to reason this through, a cannon fired upward puts it in an orbit which has the same tangential motion, but much much greater motion effectively away from the center of the Earth. (more than zero, compared to just dropping it). This increases its total momentum. More momentum for the same mass means a bigger orbit. But a bigger orbit means slower angular change over the same period of time even though its moving faster (note Pluto, way the heck out there, taking forever to go around the Sun – if you slowed its motion through the heavens you could place it closer to the sun – but its angular change on would be quicker) relative to the center of the Earth and the point at which the cannon was originally fired.

So the Earth’s angular change on its axis relative to the center and some chosen point at a chosen time is 360 degrees in 24 hours (ish), but a sufficiently powerful cannon firing that cannonball would angularly be slower. Therefore it would fall back down to the surface a hair behind.

This is just a guess without working through all the ugly math. I’d love a second guess by someone else, or proof I’m out to lunch.

Consider even more handwavingly that if the cannon were powerful enough, it could fire the ball way WAY out there stupid fast, to the point that it was almost a straight line out and almost achieving escape velocity. In the time that it was out there, ‘hanging’ just barely being pulled back by Earth’s gravity, Earth could have rotated many many times.

But the thing that bugs me is I somehow doubt that this experiment could be performed to sufficient accuracy in Galileo’s day.

3

It’s sounds rather inane to me. Actual experiments would have shown exactly the opposite. Galileo might have done some handwaving, or made some assertion to this effect, but “showing” it implies that it’s true, and it’s simply not.

4

At least according to the OP, it doesn’t say if the ball fell in the direction of the orbit or not – but the way I see it, it should actually lag behind given sufficient force perpendicular to the rotation of the Earth.

Why is it ‘simply not’? Why would actual experiments show the opposite? I’d like to know where the flaw in my reasoning is.

5

As with the ball dropped from a mast in the OP you can do the experiment yourself quite easily.

While driving down the highway drop something in your car. Bet the object will land directly under where you dropped it.

You should be glad for this too. Imagine flying in a plane at nearly 600 MPH and jumping. Thanksfully you do not hit the back of the plane at 600 MPH.

Chances are it is quite difficult to launch a cannonball perfectly straight up. Add to that wind, wind resistance and so on and you are likely to find your cannonball comes down not quite on top of its launch point. Still, if you did everything perfectly and in a vacuum the ball will go straight up and come straight back to where it was launched.

6

But the situations aren’t quite correct in your analogy. The distance from your car window to the ground is miniscule compared to, say, rocketry sized distances, or even giant-cannon distances.

Furthermore, trivial experiments like the car experiment assume all sorts of things like gravity pulling uniformly in one direction on a flat surface instead of pointing towards a single location on, say, an idealized sphere.

All of your examples are perfectly legitimate as stated only because of the extremely small distances and times involved.

I, too, doubt Galileo could have really tested this in his time, but I personally think its because of measurement error.

Are you seriously suggesting that if you punted a cannonball straight up at, say, 11 kilometers a second, exactly vertically, at the equator, just below escape velocity, that while you waited for it to come back to Earth that it would stay exactly above the point of the Earth from which it left?

7

I would think the Coriolis force would have some measurable effect on the cannonball’s trajectory. No one’s mentioned that yet.

8

Definitely get a cite. If the claim were correct as stated, then the cannonball would fly westward at several hundred miles an hour.

9

Okay, right. I disbelieve it would fall at a distance equivalent to the Earth’s surface motion. I maybe was misinterpreting the paragraph. But I still think it would fall behind the firing point by an amount that increases with the force at which it’s fired (for the reasoning I tried above). For the largest ‘cannon’ ever made, perhaps not much more than a few centimeters. Damn, I’m going to have to do the math now.

10

Yeah, that occurred to me earlier, but I forgot to include it in the OP.

This claim gets fishier and fishier the more I look at it – but I just wanted to make sure I wasn’t overlooking anything obvious. I’ll query. Thanks, guys!

(Carry on the discussion if you like!)

11

Oh well, I’m still curious. :slight_smile: Let us know what you find! :smiley:

12

Barring wind-resistence (ie, in a vacuum), yes. Why wouldn’t it?

13

Are you seriously suggesting that if you punted a cannonball straight up at, say, 11 kilometers a second, exactly vertically, at the equator, just below escape velocity, that while you waited for it to come back to Earth that it would stay exactly above the point of the Earth from which it left?

Yes, it is called Conservation of Angular Momentum (which can be derived from Newton’s Laws).

If it weren’t so, people would just get on a helicopter and hover for 24 hours for a trip around the globe.

As for the OP Scarlett, your instincts were right… Go talk with your author.

14

Fire a cannonball straight up with such force that it manages to get out to Pluto before being dragged back to Earth. Neglect other planets. Neglect that the Earth is going around the sun.

For the cannonball to stay above the same point on the surface of the earth at that distance, it has to be moving at the same angular rate. That’s 360 degrees per 24 hours, or 2*PI radians/24 hours.

At the distance of Pluto, 40 AUs from the Sun and about the same from us for all intents and purposes, this is 40149000000 kilometers out, or 5960000000 away. At this distance, a complete circle is 59600000002*PI kilometers, or 37447784430 kilometers in 24 hours. This is 433423 kilometers per second. Just to stay above the surface of the planet (which you say it is doing, unaided) at the distance of Pluto.

That’s faster than the speed of light.

I agree the Galileo experiment’s bogus inasmuch as the cannonball wouldn’t move exactly the same amount as the Earth’s surface rotates for any cannon that could be used today – it’d go up and come straight back down. But fire it far enough away, and you could have the Earth rotate thousands of times before the cannonball ever came back down.

15

The assertion is ridiculous. If the canon ball doesnt fall to the same point, for the reasons mentioned then the only thing keeping the earth from “slipping” underneath my feat is the traction in my Nikes.

16

Assuming a spherical Earth, rotating once per day, in vacuum: A canonball fired straight up would land a little west of the launching point, and a cannonball dropped from a height would land a little east of the dropping point. This effect is not large: For instance, on the equator, a cannonball dropped from a height of 100 m would be deflected by 2.2 cm. The effect is smaller off the equator, and larger for greater heights. Specifically, the deflection is [symbol]w[/symbol]g sin[symbol]q/symbol[sup]3/2[/sup]/3 ([symbol]w[/symbol] is the angular frequency of the Earth, g is the acceleration due to gravity, [symbol]q[/symbol] is the latitude, and h is the height). The westward deflection of a particle fired from the ground is twice this amount.

Did Galileo have an apparatus sensitive enough to measure that? I don’t know.

17

I’m sorry, but I’m not talking about the Earth slipping away beneath you. The assertion of the OP is ridiculous because of what you just mentioned. But the assertion that a cannonball fired away from the Earth with sufficient force maintains its position above the same point on the Earth is also ridiculous.

Unless you can show me how the cannonball, fired out as far as I suggested, manages to exceed the speed of light.

The difference?

While you are on the surface of the Earth, there is no measureable way to just ‘jump up’ sufficiently far (even if it were a vacuum) to be able to measure such a miniscule amount of distance that would result.

Again, I am not claiming the distance difference would be proportional to the rotational distance of Earth as per the OP. But there would be a difference, and it increases the further out you go. If it doesn’t, then you end up with the clearly ludicrous idea that a cannonball fired out to Pluto is exceeding the speed of light.

Or there’s another hole in my logic, but I can’t see it.

18

Although his last post has made it abundantly clear, William Ashbless’s point is correct. Maybe you’d need to perform it on a planet without atmosphere to see it clearly with a cannon, but it would happen.

Looking at it again, I think the quote from the book might simply be ambiguous. “Where it was fired” might mean relative to the earth’s center, not at the mouth of the cannon. In other words, that its inertia causes it to continue to move around the earth’s center, which as has been said is pretty much true. I’d still doubt that a physical experiment was performed by Galileo.

19

Chronos, THANK YOU.

Do you have a proof for this, just out of curiousity?

20

panamajack, thanks to you too. I thought I was going mad. :slight_smile: