Gravity is a pretty big deal. At least it excites me. It makes me think about space travel, and about the possibility of mankind having a future elsewhere after the sun can no longer support life here. In the last hundred years, science has made great advances in understanding gravity since the 1915 publication of Albert Einstein’s general theory of relativity. However, the lay person struggles not only to understand Einstein’s theory of gravity, but even to grasp the simpler Newtonian theory that Einstein improved on. This bothers me.
Up until Einstein, Isaac Newton’s theory of gravity explained any measurement of gravity that anyone had the imagination and instruments to carry out. It’s a good theory, and though it seems ancient in comparison to relativity, it contains subtleties that we have yet to appreciate. We’re still somewhat stuck on the even older ideas of Galileo Galilei. Galileo, you will recall, is famously said to have disproven Aristotle’s teaching that heavier objects ought to fall faster than lighter ones by dropping balls from the Tower of Pisa and watching what happened rather than merely trying to reason out what ought to happen in a sensible world. As I will show in part two of this series, Galileo’s findings were not the final word on the question of whether heavier objects fall faster than lighter ones, and the answer will take most people by surprise.
I will present three principles that are commonly taught on the subject of gravity, then criticize each of them, and show how Newton’s theory of gravity contradicts them. Then I will discuss the problems common to these three misconceptions.
Galileo demonstrated that the downward pull of gravity acts without regard to an object’s horizontal motion: an object dropped from a certain height will accelerate toward the Earth no faster or slower than an object thrown horizontally from that same height. Likewise, the object’s horizontal motion is unaffected by gravity: an object thrown horizontally will continue with the same horizontal speed until it falls to the ground. Even if an object is thrown in a direction having both horizontal and vertical aspects, the horizontal aspect of its motion will remain constant as long as the object is in flight, while the vertical aspect of its motion will steadily gain downward speed (or lose upward speed . . . to a scientist, that’s the same thing).
In free fall, horizontal motion is uniform, meaning that position varies in direct proportion to elapsed time; at one meter per second, an object will move horizontally two meters in two seconds, and so on. Vertical position is accelerated, meaning that velocity varies in proportion to elapsed time; after being dropped from a state of rest, the object will gain roughly 10 meters per second . . . per second. After one second it will be falling at 10 meters per second; after two seconds, it falls at 20 meters per second. This means that vertical position will vary in direct proportion to elapsed time squared.
As an aside: In the discussion that follows, I will ignore the cases in which an object goes straight up and/or straight down. For one thing, can anyone be sure that something is going in an exactly vertical direction? And besides that, a straight line is boring.
The demonstrated independence of the downward acceleration and the uniform horizontal motion suggests that every projectile will follow a general shape called a parabola. A parabola is an open curve that is round in the middle and has symmetric sides that become increasingly steep as they extend away from the middle, almost becoming parallel to one another. You may recall that the parabola is a function of two variables (we’ll represent the variable horizontal and vertical directions with the letters x and y), which in its simplest and most familiar form is written as y=x2 . Math aside, most everyone develops a natural ability to compute a parabolic arc in their head, whether it’s to catch a football or toss a shirt into the laundry basket.
Illustration: The parabola y=x2. The trajectory of a thrown object resembles the upside-down form of this shape.
In the domain of everyday human activity, the parabola is a fine description of the path of falling objects. Over small distances, we would be unable to measure any deviation from this parabolic trajectory (or at the very least we could account for any deviation as being caused by gusts of wind or some other force). On a large scale, however, it is immediately apparent that the open curve of the parabola does not describe the motion of satellites around the Earth, nor of the planets around the sun. These objects follow a path that appears closed, yet "falling" is precisely what those objects are doing. You may ask, what keeps a satellite from falling to earth? The satellite is going so fast horizontally that it keeps missing. It goes around the Earth in a shape called an ellipse. Likewise, the planets all have elliptical orbits around the sun. One of the great revelations of Newton’s theory of gravity is that it showed that objects on the Earth follow the same laws as objects in the sky. As above, so below.
Illustration: An ellipse is one of the shapes obtained by intersecting a cone with a plane.
Why then, according to Newton, might we expect an earthly projectile to follow an elliptical path rather than a parabolic one? First, because the pull of gravity is weaker at greater distances from the Earth. An object at the top its arc will accelerate more slowly than it does at the bottom. Second and more obviously, the path is elliptical because the Earth is round and the projectile always accelerates towards the center of the Earth; therefore the "downward" as measured from the beginning of the projectile’s flight may be in a different direction than the "downward" that is at the other end. Remember that the parabolic trajectory defined by Galileo is dependent on the conditions of motion which he observed. These two conditions which Newton took into account and Galileo did not result in the elliptical trajectory.
That being so, why do we not in fact see projectiles follow an elliptical path? Actually, we do. Over small distances, a parabolic path will closely match an elliptical one, and we typically deal with distances that are small enough that we can’t measure the difference. This near match reflects the fact that in small spaces, gravity is not measurably different between the top and bottom, and that in these spaces, the Earth is so nearly flat that "downward" from any point is in the same direction as from any other point, at least as accurately as we are able to measure. The other reason that the path does not appear obviously elliptical is that it typically intersects the Earth at some point, interrupting the projectile’s flight before it can curve back around on itself.
"So . . . ," I hear some of you saying, "the parabola is out. Gravity causes objects to follow the shape of an ellipse. Got it." Not exactly; there’s more to this story.
Newton imagined what would happen to a cannonball if it were fired from a very high mountaintop at greater and greater velocities, achieving ever greater distances, until it ultimately went into orbit. We will explore the same scenario.
At speeds that would be considered possible for a cannonball, the ball would follow an elliptical path. At speeds close to zero, it would drop nearly in a straight line, forming a small segment of an ellipse that would be "squeezed in" from side to side until the two sides nearly met. The faster the ball flew, the further away it would strike the ground, and the more the ellipse would open up. At more extreme speeds, the ball would actually disappear over the horizon.
The ellipse is much easier to visualize if we pack the Earth into a tiny little ball and leave the cannon way up there on its own, high above the surface. At speeds near zero, the ball rolls lazily out the barrel and drops nearly straight down, just as we imagined before, with just a little motion to the right. But this time, instead of striking an enormous Earth just below it, it keeps falling toward the tiny, compacted, far-way Earth . . . faster and faster . . . and it misses, just barely. It shoots by, but then goes slower and slower, because now the Earth is pulling it back. It keeps going, barely reaching a point on the opposite side of the Earth‐which is just as far away from the ground as it was when it was fired‐and now having that same lazy speed that it had coming out of the barrel, just going a little to the left now. And the process repeats in the opposite direction with the ball ending up with the same position and velocity as when it started, completing one elliptical orbit that leads directly into another. If we pull back far enough in our perspective, the elliptical orbit of the cannonball is so narrow as to give the appearance of being a straight line.
The faster the cannonball is fired, the more widely it misses the tiny, compacted Earth below it, and the more rounded the elliptical orbit is. For any given velocity, the path is the same shape as the path around the full-size Earth; the size of the Earth determines only whether the ball will strike the Earth before completing its orbit.
If we continue firing the cannonball faster and faster, the ellipse eventually widens to a perfect circle (a circle is part of the ellipse family); even if the cannon is fired near ground level (on a full-size Earth) it will not strike the ground but will circle around the Earth to the place it began, again and again, maintaining the same height above ground at all times. If the ball is fired from a given altitude, this circular orbit will happen only at one particular speed.
Illustration: A cannonball is fired at successively greater speeds, taking trajectories A and B to fall to earth, circular orbit C, elliptical orbit D, and at escape velocity on trajectory E. Trajectories A and B would be more easily seen as elliptical if they did not intersect the ground.
At speeds higher than needed for a circular orbit, the cannonball‐still being fired horizontally‐will not drop but will actually gain altitude as it travels toward the opposite side of the Earth, then drop again as it returns to its starting point. This path will also be elliptical. As the cannonball is fired with greater and greater speed, the altitudes it reaches at the opposite end of this elliptical path become higher and higher. It takes longer and longer for the cannonball to return to its starting point, until another critical velocity is reached and the ball simply does not come back at all, leaving the Earth behind for good. This is called the escape velocity, and the path of the ball is now a parabola. There is one escape velocity for any given altitude around the Earth. If the cannonball is fired at escape velocity toward the ground it will of course fall to earth; but if it is fired away from or even tangent to the Earth’s surface, it will not.
At even greater speeds, the ball follows a trajectory called a hyperbola. The ellipse, circle, parabola, and hyperbola are collectively known as the conic sections, meaning that all of these shapes can be found by slicing a plane through a cone. Each of these shapes is obtained by changing the angle at which the plane meets the cone, and they are found in the same order as we discussed in increasing the speed of our imaginary projectile. The initial speed of the projectile corresponds to the angle of the plane. I intend to explore this conic-sectional model of gravity further in a future article or video.
So there you have the full picture of the geometry of Newtonian gravity. When your physics professor and your textbook tell you (and they most likely will tell you) that projectiles take a parabolic path as they fall to earth, take this with a grain of salt. After further study and/or questioning, both of them will admit that this is only very nearly true under a limited set of circumstances.