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I was explaining to my 8 year old daughter that objects in free fall follow an elliptical trajectory instead of the commonly believed parabolic one (source: https://www.forbes.com/sites/startswithabang/2020/03/12/we-all-learned-physics-biggest-myth-that-projectiles-make-a-parabola/). I told her that only on a flat Earth would an object on free fall follow a parabolic trajectory. Then she asked me what shape the Earth would have to be for an object in free fall to follow a straight line trajectory. Is it even possible?

Qmechanic
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marconi
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    If you just drop something it's in freefall and will fall straight down, so are you asking if there is a shape for which all possible trajectories, regardless of initial motion and direction, are straight lines ? – StephenG - Help Ukraine Oct 21 '21 at 21:34
  • @StephenG that seems to be what he is asking. – hft Oct 21 '21 at 21:51
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    @marconi, you might also want to explain to your 8 year old daughter that the linked-to Forbes article has a very clickbaity title. It is not a "myth" that a projectile makes a parabola, it is an approximately true statement under common circumstances where the force is (approximately) constant. This is the case for much human activity near the surface of the earth. It would be absurd to try to take into account the curvature of the earth in all projectile calculations (especially the pedagogical ones). – hft Oct 21 '21 at 22:02
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    @hft clickbaity title aside, I think the article is sound, and it explains that the difference is very tiny indeed and that it would be a parabola on a flat Earth. My daughter found the flat-Earth implication hilarious. – marconi Oct 21 '21 at 22:31
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    Frankly, I think the article is not as instructive as it sneers. It says little about the eccentricity of the orbit being so close to 1, the parabola limit, and its dependence on the escape velocity. Light, which is very light (only its energy counts) escapes in virtually straight lines. – Cosmas Zachos Oct 21 '21 at 22:40
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    @marconi Hopefully you are right that the article is better than its clickbait title. I took a quick look and it seems to double down right away about this purported "myth." Not an encouraging start. We should not describe a useful approximation as a "myth." (BTW, there are many other ways that a projectile trajectory can deviate from parabolic. For example, if you take into account non-linear air resistance, the trajectory is not a parabola. For example, if the projectile hits a droplet of water or sleet it will deviate from a "perfect" parabola (or perfect ellipse for that matter). Etc etc.) – hft Oct 21 '21 at 22:54
  • Wouldn't any shape with inversion symmetry about a line do the job? I'm not sure I'm expressing that correctly. For example, two right circular cylinders that have been truncated by identical chords and then stuck together. Is there a name for that symmetry? – garyp Oct 22 '21 at 01:41
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    There are many problems with the article linked from Forbes. For starters, the author acts as though all orbits are elliptical, which is simply not true—Newtonian gravity also allows for parabolic and hyperbolic orbits. They then use the term aphelion when they should say apogee. However, the biggest problem is that they act as though the curvature is significant at the everyday scale, saying "[e]ven over distances of just a few meters, the difference between a perfectly flat Earth and a curved Earth comes into play," despite the fact that air resistance has a much larger effect at that scale. – Sandejo Oct 22 '21 at 05:20
  • @hft I think that most people (OK, most of those that have an idea about it to begin with) would answer "parabola" without hesitation. Are there junior high or high school textbooks which mention that it's only an approximation? As an example which looks very serious and very typical, look at https://www.texasgateway.org/resource/53-projectile-motion. I hope I didn't miss it in the fine print but the page does not contain the string "elli"... The text follows the typical "trajectory": Motion can be viewed as addition of independent vectors; one of them is perpendicular to the other... – Peter - Reinstate Monica Oct 22 '21 at 06:42
  • do realize that truely parabolic trajectories are also possible in real-world gravitational fields – lineage Oct 22 '21 at 11:38
  • In the general relativistic sense, all objects can be said free fall in [https://en.wikipedia.org/wiki/Geodesics_in_general_relativity](straight lines). Not that I am recommending this become part of the conversation with a young child. – Brian B Oct 22 '21 at 16:31
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    There has to be something missing from this question. A straight line is a special case of both parabolas and ellipses, no? And doesn’t really depend on the shape of the earth. – Beanluc Oct 23 '21 at 05:47
  • An object wouldn't travel in a parabola on a flat Earth. On an infinite plane, gravity would not vary inversely to the square of the distance from the center of mass of the Earth, but it would still vary inversely to the distance, so the acceleration would decrease as its altitude increases. – Acccumulation Oct 24 '21 at 01:44
  • Closely related: https://physics.stackexchange.com/a/373256/44126 – rob Oct 24 '21 at 16:08

7 Answers7

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Then she asked me what shape the Earth would have to be for an object in free fall to follow a straight line trajectory. Is it even possible?

Yes, it is possible, under very strange (effectively purely hypothetical) circumstances.

Suppose that the shape of the earth was a uniform-density hollow spherical shell and suppose that instead of living on the outside of the earth, we lived on the inside of the shell. In this case the trajectory of a projectile will be a straight line.

The reason there is a straight line trajectory in this case is because in this case there is no gravitational force on the projectile (since there is no mass within the inner part of the sphere and since the force from the shell conveniently happens to exactly cancel everywhere within the shell).

hft
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  • Why is this getting a downvote? Explain yourself. – hft Oct 21 '21 at 21:53
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    I didn't downvote or upvote, but why do you need the shell at all in this example? This is essentially equivalent to saying "things move in a straight line if there's no gravitational field", since trajectories will generally be straight lines outside of the shell. – Andrew Oct 21 '21 at 22:20
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    The daughter asked for an example earth. This seems like a more fun example than proposing that the example earth is no earth at all. – hft Oct 21 '21 at 22:55
  • This might be even more difficult to explain to a young girl as it's immediately going to occur to her to ask "why there's no gravity inside the shell ?". It's hard enough for physcs undergrads to grasp this. Otherwise it's a legitimate answer. – StephenG - Help Ukraine Oct 21 '21 at 23:55
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    I upvoted because this is a non-trivial mass that fits the criteria. My first thought was there couldn't be one. I agree that this is beyond an 8 year old explanation. But other readers are beyond 8 years old. – mmesser314 Oct 22 '21 at 02:39
  • If you jump from one side of the shell, do you need to wait and cross the 12000km? – Eric Duminil Oct 22 '21 at 08:19
  • @StephenG It's never too early to learn your kids about calculus and Newton's shell theorem ;) – AccidentalTaylorExpansion Oct 22 '21 at 11:05
  • This answer seems to gloss over the gravity exerted by the shell itself, which would be "upward" (i.e. away from the center), specifically when the shell is not uniformly distributed. However, interestingly, for a uniformly distributed shell, the gravitational forces in all directions (since the shell is all around you) actually cancel each other out. While the part of the shell you're closer to attracts you slightly more than the other side, there is more shell on the other side which (per unit) exerts a smaller force. It's a zero-sum game. [..] – Flater Oct 22 '21 at 13:09
  • [..] This means that if you e.g. tunnel 50km down into a planet and stand there, you experience the same gravity as if you were standing on the surface of a planet with a 50km smaller radius. Any planetary mass above you (= further from the center than you are, in all directions) can be ignored. This assumes that the planetary mass is uniformly distributed and ignores the mass different due to the tunnel you dug. – Flater Oct 22 '21 at 13:10
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    @Flater: It doesn't seem to gloss over it at all. It specifically says a uniform density spherical shell... – Chris Oct 22 '21 at 14:44
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    Can an object in true zero gravity even be said to be in free fall? Free fall is defined as motion where gravity is the only acting force. If there is no gravity, there are no acting forces at all - the object is "free", but it's definitely not "falling". – Nuclear Hoagie Oct 22 '21 at 16:10
  • @NuclearHoagie Sure; define free fall to be motion along a geodesic in spacetime, whatever its shape might be, and there you go. No need to invoke a special case for "no gravity". – Filip Milovanović Oct 23 '21 at 11:15
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    A hollow cylindrical earth would cause an object with an initial impetus along its axis to oscillate along that axis, even if it exceeded the physical length of the cylinder. In fact, any body with rotational symmetry and a hole through it should work. – Paul_Pedant Oct 23 '21 at 21:21
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An object will follow a straight line trajectory if acceleration is in the direction of its velocity. In a gravitational field, acceleration is in a fixed direction. The trajectory will be straight only of velocity is in that direction. On a spherical planet, it will be straight if the initial velocity is straight up or down.

There is no gravitational field that can make the trajectory straight given an arbitrary initial velocity. (Except a field that is $0$ everywhere.)

mmesser314
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  • So the answer is if the Earth didn't exist. – user253751 Oct 22 '21 at 09:16
  • @user253751 - That is one answer. Another is if Earth was hollow and you were inside. As other answers show, that is another way to get $0$ gravity. – mmesser314 Oct 22 '21 at 14:15
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    A nitpick would say that no gravitational field at all (except the one created by the ball itself, whose net force on the ball is zero) would imply that there is nothing at all. No Earth, but also no Sun, no Jupiter, no Alpha Centauri, no Andromeda, no GN-z11. A hollow Shell only cancels out the Shell's own gravitation; it does not shield from external influence (it is no "gravitational Faraday Cage"). And that doesn't even consider the spacetime geometry or the question what exactly, in GR, you would consider a "straight line". – Peter - Reinstate Monica Oct 22 '21 at 14:29
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If you mean straight lines on the spherical earth's surface, then objects in free-fall already follow straight line trajectories (if they are a given an initial velocity in the direction of the acceleration or just dropped from a height with no sideways components of velocity). Since the Earth is spherical, the strength of the gravitational force varies with the distance to the center of the Earth according to $${\bf g}=\frac{GM}{r^2}{\bf \hat r}$$ where ${\bf\hat r}$ points to the center of the earth. The path an object takes in free-fall is perpendicular to a tangent line on the earth’s surface.

I told her that only on a flat Earth would an object on free fall follow a parabolic trajectory.

I doubt that on a flat earth objects would follow parabolic trajectories. If we assumed that the earth was somehow shaped like a flat disc, the gravitational force would be greatest in the center of the disc and objects would free fall toward the surface in straight lines only at the center of this disc.

As you moved further from this center, gravity would pull more and more horizontally toward this center, so that an object dropped at the rim of this disc could fall diagonally (or almost horizontally depending on how large the radius of this disc is). Straight downward free-fall motion would be possible at the center of the disk only.

Then she asked me what shape the Earth would have to be for an object in free fall to follow a straight line trajectory. Is it even possible?

If you mean a straight-line horizontal trajectory (parallel to the ground) then if the earth was a very large flat disc, as stated above the gravitational force would point to the center of the disc. Given that the disc is very large, at the outer regions of the disc, if you were to drop an object, it would move (almost) in such a straight horizontal line.

joseph h
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    if by 'flat earth', he meant an infinite plane of mass, which does have a constant gravitational field, and produces parabolic trajectories – Nyra Oct 23 '21 at 04:09
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I like the shell answer, as it is not trivial. The only other answer is the trivial one: no shape, as in no Earth.

Now if she wants a straight line with non-zero acceleration, then she is out of luck.

JEB
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Imagine the transformation of a spherical Earth into a flat Earth. In what directions is the distortion taking place? What would it look like if the distortion continued in the same directions? That's right, the surface would fold upward into a bowl (with the original outer surface of the Earth on the inside) then a bottle, then a sphere -- the hollow Earth of hft's answer.

This should be easy for an 8-year-old to visualize. The point here is that the idea of a hypothetical hollow Earth doesn't come out of nowhere.

Now imagine the transformation of an elliptical fall into a parabolic fall. How are they different? What changes to make an ellipse into a parabola? Can that distortion be continued in the same directions?

Aside from the fact that the ellipse is closed and the parabola open, a parabolic arc is visually flatter than an elliptic arc. Inquiring 8-year-old minds want to know: how flat can it get?

A. I. Breveleri
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It's been mentioned in answers already, but I just want to reinforce that in terms of an opportunity for teaching a young child about physics, the following answer is actually a very good one: it would travel in a straight line if there were no gravity at all, for example if there were no Earth at all. In fact, "free falling" objects in deep space, far away from any planets, do in fact move in straight lines.

This of course is Newton's first law of motion: an object will stay still or keep moving in a straight line if no force acts on it. So this is a great opportunity to teach a really foundational principle of physics.

(I used the words "free falling" above. One could argue that if objects in deep space don't experience a gravitational force then they are not really "falling" - but the point is that moving freely under no gravitational force is just a limiting case of "falling", namely the one in which the gravitational field is zero.)

N. Virgo
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  • Just curious; where are "free falling" objects (in deep space, far away from any planets) free-falling to? Is it reasonable to call such a movement a fall? – Caius Jard Oct 22 '21 at 16:29
  • @CaiusJard why do you think "fall" requires a vector? – fectin Oct 22 '21 at 16:35
  • @fectin can you rephrase the question? – Caius Jard Oct 22 '21 at 16:48
  • @CaiusJard your original question doesn't really make sense. You ask about where something is falling "to" as part of picking on the "fall" part of free-fall. But why do you think the word "fall" requires a direction in the first place? – fectin Oct 22 '21 at 16:58
  • What would "fall" mean if it didn't have a direction? If it's moving, it has a direction. If it's not moving...it's not falling. – Beska Oct 22 '21 at 17:33
  • @fectin because "[free] falling" is generally understood to be an acceleration due to gravity, and an acceleration is a vector. In simple terms, objects "fall" towards the object whose gravity they are experiencing. A ball falls to earth, the earth falls to(ward) the sun.. I'm trying to get some clarity on the scenario presented in the answer, of objects in deep space (presumably experiencing no gravity) that are free-falling; what acceleration do they experience and in what direction "if there were no gravity at all". – Caius Jard Oct 22 '21 at 17:46
  • @CaiusJard sorry for the delayed reply. Objects in deep space experience a gravitational acceleration of zero. You can argue, quite reasonably, that if there's no gravitational acceleration then they are not falling. I used "free-falling" in the answer to emphasise that the situation of no acceleration is really just a special case of falling in a gravitational field (it is the special case in which the gravitational field is zero) but I can see that could be confusing - I will try to fix the answer. – N. Virgo Oct 24 '21 at 10:05
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An object in free fall is accelerated towards the centre of gravity of the Earth. That means that its trajectory will be a straight line if and only if , it has no component of velocity normal to the line between it and the centre of the Earth. Off the top of my head, I imagine that to hold true regardless of what shape the Earth takes.

Marco Ocram
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    It does depend on the shape of the Earth: in the far field any shape would indeed look like a point mass, but as you come closer, local bulges will tend to curve free fall trajectories towards them. – Ruslan Oct 22 '21 at 14:50