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Yesterday, I understood what it means to say that the moon is constantly falling (from a lecture by Richard Feynman). In the picture below there is the moon in green which is orbiting the earth in grey. Now the moon wants to go at a tangent and travel along the arrow coming out of it. Say after one second it arrives at the red disc. Due to gravity it falls down toward the earth and ends up at the blue disc. The amount that it falls makes it reach the orbital path. So the moon is constantly falling into the orbital path, which is what makes it orbit.

enter image description here

The trouble I'm having is: shouldn't the amount of "fall" travelled by the moon increase over time? The moon's speed toward the earth accelerates but its tangential velocity is constant. So how can the two velocities stay in balance? This model assumes that the moon will always fall the same distance every second.

So is the model wrong or am I missing something?

Extra points to whoever explains: how come when you do the calculation that Feynman does in the lecture, to find the acceleration due to gravity on earth's surface, you get half the acceleration you're supposed to get (Feynman says that the acceleration is $16 ~\mathrm{ft}/\mathrm{s}^2$, but it's actually twice that).

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Short answer: he was/is correct. Long (and more interesting!) answers: see below. –  user12345 Mar 30 '13 at 20:17
    
Just to add in case it helps: OP wrote "The moon's speed toward the earth accelerates but its tangential velocity is constant". The tangential speed is constant. If you like, all of the acceleration towards the Earth in each little step goes into changing the direction of the Moon's velocity, keeping it perpendicular to the direction of the acceleration. This means that the speed of the Moon towards the Earth never "accelerates", actually it is always zero. –  Mark Mitchison Mar 30 '13 at 21:22
    
"shouldn't the amount of "fall" travelled by the moon increase over time?" - no? The moon is being accelerated downwards, but because it's going forward so fast, all the previously accumulated velocity is now in the forwards direction. "The moon's speed toward the earth accelerates" - This is not true. You need to realise that the direction towards Earth, 'downwards', is constantly changing. –  Joren Mar 31 '13 at 0:07
    
I think the 16ft/s^2 must be a misprint or lost factor of 2. Feynman could also have "slips of attention". –  anna v Mar 31 '13 at 4:46
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@mtanti "Feynman says that the acceleration is $16 ~\mathrm{ft}/\mathrm{s}^2$ […]" When? I hear him say that things fall $16$ feet in a second, which fits nicely with an acceleration of $32 ~\mathrm{ft}/\mathrm{s}^2$ and a starting speed of $0~\mathrm{ft}/\mathrm{s}$. –  Glen The Udderboat Mar 31 '13 at 12:52
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4 Answers 4

up vote 16 down vote accepted

What's actually happening is something more like this:

Approximation of circular orbital motion using the velocity Verlet method

Here, $x_0$ and $v_0$ are the initial position and velocity of the moon, $a_0$ is the acceleration experienced by the moon due to gravity at $x_0$, and $\Delta t$ is a small time step.

In the absence of gravity, the moon would travel at the constant velocity $v_0$, and would thus move a distance of $v_0 \Delta t$ during the first time step, as shown by the arrow from the green circle to the red one. However, as it moves, the moon is also falling under gravity. Thus, the actual distance it travels, assuming the gravitational acceleration stays approximately constant, is $v_0 \Delta t + \frac12 a_0 \Delta t^2$ plus some higher-order terms caused by the change in the acceleration over time, which I'll neglect.

However, moon's velocity is also changing due to gravity. Assuming that the change in the gravitational acceleration is approximately linear, the new velocity of the moon, when it's at the blue circle marking its new position $x_1$ after the first time step, is $v_1 = v_0 + \frac12(a_0 + a_1)\Delta t$. Thus, after the first time step, the moon is no longer moving horizontally towards the gray circle, but again along the circle's tangent towards the spot marked with the second red circle.

Over the second time step, the moon again starts off moving towards the next red circle, but falls down to the blue circle due to gravity. In the process, its velocity also changes, so that it's now moving towards the third red circle, and so on.

The key thing to note is that, as the moon moves along its circular path, the acceleration due to gravity is always orthogonal to the moon's velocity. Thus, while the moon's velocity vector changes, its magnitude does not.


Ps. Of course, the picture I drew and described above, with its discrete time steps, is just an approximation of the true physics, where the position, velocity and acceleration of the moon all change continuously over time. While it is indeed a valid approximation, in the sense that we recover the correct differential equations of motion from it if we take the limit as $\Delta t$ tends towards zero, it's in that sense no more or less valid than any other such approximation, of which there are infinitely many.

However, I didn't just pull the particular approximation I showed above out of a hat. I chose it because it actually corresponds to a very nice method of numerically solving such equations of motion, known as the velocity Verlet method. The neat thing about the Verlet method is that it's a symplectic integrator, meaning that it conserves a quantity approximating the total energy of the system. In particular, this means that, if we use the velocity Verlet approximation to simulate the motion of the moon, it actually will stay in a stable orbit even if the time step is rather large, as it is in the picture above.

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+1 for the picture. How did you do it? –  becko Mar 31 '13 at 2:42
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@becko: I drew it in GeoGebra. Here's the worksheet file, if you want to play with it. (Try dragging the $x_1$ point around!) –  Ilmari Karonen Mar 31 '13 at 2:45
    
I'm not sure if I missed anything in your explanation, but are you saying that the moon speeds up its tangential velocity? And why should a1 affect the velocity v1 if it hasn't fallen by a1 yet? –  mtanti Mar 31 '13 at 10:16
    
@mtanti: The tangential velocity stays constant, as you can see from the picture. As for why $v_1$ should depend on $a_1$, remember that the acceleration vector is changing continuously from $a_0$ to $a_1$ as the moon moves from $x_0$ to $x_1$. Taking the mean of $a_0$ and $a_1$ in the approximate formula for $v_1$ is just a linear approximation of this gradual change. –  Ilmari Karonen Mar 31 '13 at 18:05
    
I think I finally got it when I wrote a program which simulates this. The tangential velocity does speed up as it falls since the new tangential velocity will be the resultant vector (adding the previous tangential velocity to the falling velocity). This makes the moon able to speed past the earth, although not in a circular orbit.In my simulations the moon would always crash into the earth eventually but maybe its because I didn't take into account other factors like centrifugal force. –  mtanti May 4 '13 at 16:49
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Picture this. Take your desk and put in top of the tallest building on earth. Put a penny on the desk and flick it with your finger. (The point of this is that we are only giving the penny a horizontal initial velocity and letting gravity do the rest for us, that is we are letting the 'falling' do all the work.) The penny falls and hits the ground. Now flick it a bit harder and it will go a bit further before it hits the ground. Now hit much harder. It will go much further, and if you hit it hard enough, you will get a bit of extra time in the air since the earth's curvature will start to come into play. Note that the penny is still falling in all cases. Now just keep flicking the penny until eventually you put it in orbit. That is, orbiting is just a special case of 'falling'.

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I'm sorry but I've been seeing this Newton's cannon ball thought experiment for ages and I never quite see why it should eventually never hit the ground. The curvature towards the earth should increase over time as it is accelerating and so I don't see why it shouldn't spiral into the ground. –  mtanti Mar 30 '13 at 19:00
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Strap the penny to a brick of C4. But carefully so that the penny only gets a horizontal velocity when you blow up the C4. In principle the penny can certainly escape the earth even though you are only giving it an horizontal velocity. Does this make anymore sense? –  DJBunk Mar 30 '13 at 19:02
    
@mtanti The moon is not accellerating its fall! The accelleration is constant, and its direction is always towards the centre of the Earth. This means that, if you give the penny a certain amount of horizontal velocity, then it will do a complete rotation around the Earth and return to the inital position with the same velocity, which implies that it will do an other round in exactly the same way(assuming time and space are isotropic, and if they aren't we are pretty much screwed). –  Bakuriu Mar 30 '13 at 22:07
    
@mtanti Also, an other thing, remember that the acceleration changes its direction, which means that at some point the penny increases by x m/s on the x axis, but after some time it decreases by the same amount(if you give it enough velocity to complete a round). –  Bakuriu Mar 30 '13 at 22:13
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This doesn't answer the question. –  Larry Harson Mar 30 '13 at 22:32
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Assume the distance between the green and red balls is very short. The Moon's horizontal speed happens to be such that as it travels horizontally from the green to the red, it falls to the blue (this is what we're assuming in the first place, after all).

Now the Moon has traveled horizontally and fallen, so that it's at the same distance from the Earth as when it started.

We've just gone from step 0 to step 1.

Now redraw the diagram by rotating it slightly. The Moon is now at the position of the new, redrawn green ball, with the same velocity it had at step 0. It's going to do exactly the same thing going from step 1 to step 2, and 2 to 3, and so forth. This process repeats itself infinitesimally and continuously.

If the Moon's velocity were different, if it were moving faster or slower, or in a direction other than perpendicular to the line connecting the Moon and the Earth, then it wouldn't behave this way. Its velocity happens to be very close, in both direction and magnitude (speed) to keep it at the same distance from the Earth with a constant speed. And if it maintains a constant distance and speed over a short period of time, it will continue to do so indefinitely.

Note that this description doesn't make use of the fact that gravity falls off as the square of the distance; since the distance remains constant, all we need to assume is that it's constant for a given distance, and radially symmetric. In reality, the Moon's orbit is distinctly elliptical, something that's possible only with an inverse square force. If the gravitational force fell of as, say, the cube of the distance, then an elliptical orbit would not be stable -- but a perfectly circular orbit still would be, at least in the absence of any perturbations.

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Pedantic comment: "stable" meaning "not doomed" rather than the usual "stable against small perturbations". Nice answer, though. +1 –  Mark Mitchison Mar 30 '13 at 21:33
    
@MarkMitchison: Good point, I've updated my answer accordingly. –  Keith Thompson Mar 30 '13 at 21:55
    
I get everything you said but my question was: shouldn't the distance the ball travels toward the earth during the next iteration be larger since it is accelerating? –  mtanti Mar 31 '13 at 2:23
    
@mtanti: If you stay in the original frame of reference, the distance the ball moves "down" is larger. But the Earth is no longer straight "down"; it's now below and a little bit "behind" the Moon's new position. Over a short time span, the Moon's motion approximates the parabola it would follow in a uniform gravitational field; the Earth is at the center of the circle that's tangent to that parabola. –  Keith Thompson Mar 31 '13 at 6:27
    
@mtanti The point here is that the moon does not speed up. It is accelerating because its direction is changing. A change of velocity can be a change in speed, or a change in direction, or both. –  Harry Weston Apr 3 '13 at 19:13
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Your diagram of the red and green balls will make for a great explanation. It seems your hang-up is that you'd expect the green ball's velocity to get faster and faster.

You can imagine a few scenarios that should make it more clear:

Scenario 1: Let go of the green ball with no horizontal velocity. It's velocity will increase as it falls towards the center.

Scenario 2: Imagine the green ball moving away from the center rapidly but still with no horizontal velocity. Instead of falling towards the center, the acceleration of gravity must slow it down first before reversing its direction. In this case it actually starts out getting slower and slower.

Scenario 3: Give the green ball horizontal velocity like in the diagram. As it moves towards the red position it is getting farther away from the center like in scenario 2. To get it to fall into the center it would need to be slowed down before it could be sped up. In the process of slowing it down by moving it from where it was aimed at (the red position) to where it actually ends up (the blue position) the ball speeds up perfectly balancing out the amount it needed to be slowed down.

That is, the reason something doesn't fall faster and faster is that orbit is the perfect balance between needing to slow down and speed up the object before it can fall towards the center. It's the perfect mix between scenario 1 and 2.

If you're trying to imagine this discrete stair-step-like orbit remember that it doesn't happen in discrete steps. Acceleration is happening on the ball constantly (infinitely often) and all of the acceleration that would be used to speed up the ball is instead used just to deflect its trajectory, thereby keeping the absolute magnitude of its velocity constant.

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So what you're saying is that as the moon gets out of orbit due to its tangential velocity, it decelerates and thus returns to it's original speed before it falls back into orbit? In which case, the orbit consists of infinitesimal wiggles of slowing down and speed up. –  mtanti Mar 31 '13 at 2:27
    
@mtanti That's one way to look at it however remember this is happening continuously so the magnitude of the velocity is truly constant. Just as integrating under a curve can be thought of as the sum of rectangles under the curve but because it's done infinitely instead of a stair-step approximation the curve is perfectly smooth. –  Brandon Enright Mar 31 '13 at 4:02
    
Wait, why should the fact that this happens continuously have to do with the speed being constant? –  mtanti Mar 31 '13 at 11:48
    
@mtanti See my comment above. The moon's speed doesn't change, only its direction. The effect of gravity is to change its velocity by altering its direction, not by altering its speed. –  Harry Weston Apr 3 '13 at 19:18
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