What shape the Earth would have to be for an object in free fall to follow a straight line trajectory?

Question

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?

Answer 1

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).

Answer 2

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.)

Answer 3

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.

Answer 4

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.

Answer 5

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?

Answer 6

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.)

Answer 7

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.