Dave Computes Orbits and Projectile Motion

Part 1:  It's not a parabola.

When I'm standing on Earth and I throw a projectile in any direction and at any speed (less than escape velocity), most knowledgeable calculus or physics students will say that the projectile moves in a parabolic path. But that assumes that there is a constant downward acceleration.

What kind of path does the projectile follow when we take into account that the acceleration at its highest point is slightly less than the acceleration at the surface of the earth, and that the direction of that acceleration is always toward the center of the earth? The answer is surprisingly simple and illuminating.

Picture a satellite (of negligible mass) in orbit around the earth, moving in an elongated elliptical orbit:

(All diagrams in this post are the results of computer simulations.) We know that the satellite's acceleration depends only on the mass of the Earth and the distance to Earth's center. Thus, increasing the size of the Earth without changing its mass would have no effect on the satellite's orbit:

Things change only if the Earth grows so large that its surface intersects with the satellite's path:

Now, of course, the "satellite" crashes. When the Earth obscures enough of the elliptical path, the object is now more properly called a projectile, and the path appears to be parabolic:

So there's our answer. Projectiles, like satellites, follow elliptical paths. Always. Let's take that in for a moment. It means every time we throw a ball, we're really launching it into orbit around the center of the Earth--until the ground or other obstacle interrupts that orbit. Even a jumping human is briefly in orbit.


Part 2:  But it could be a parabola.

Imagine that an object above Earth's surface is given a horizontal velocity and then allowed to orbit the Earth in an elliptical path, as shown below. (The object's initial position is shown at the top of the orbit in each diagram.) Recall that an ellipse has two foci points. One focus is at the center of the Earth. If the initial velocity is relatively small, then the other focus is high above Earth's surface, just below the object's starting position.

If the object is given a larger initial velocity (still entirely in the horizontal direction), then the elliptical orbit grows wider and the second focus point moves closer to the Earth.

With a large enough initial velocity, the two foci meet in the center of the Earth, and a circular orbit occurs:

With an even larger initial velocity, the second foci moves to the other side of the Earth:

A larger initial velocity moves the focus further down in the diagram, and the orbit becomes longer and more eccentric.

With greater and greater initial velocities, the second focus moves as far down in the diagram as we wish, and the object moves as far away from Earth as we like, only to return again along its elliptical orbit.

At a certain high but finite velocity, the second focus point moves infinitely far away, and the object's path is no longer elliptical. Now the path is a parabola, and the object will escape Earth's gravity. Along this path, the object slows down more and more as it gets further and further from Earth, coming to a complete stop only at an imaginary point infinitely far away.

An even larger initial velocity now results in a hyperbolic path. Again, the object will never return, but now it will never slow down completely either. The second focus of the hyperbola now appears above the starting position, as if it wrapped around from infinity.

Neglecting relativistic effects (as I have throughout this post), in the limit, an arbitrarily large initial velocity will result in a horizontal path.

Part 3:  Further Thoughts

Barring velocities high enough to escape Earth's gravity, we've seen that an initially horizontal velocity always results in an elliptical path. But what about initial velocities with nonzero vertical components? No matter how its path starts, the object eventually reaches a point where its velocity is purely tangential, and that point will be the furthest or nearest point in an elliptical orbit aligned at some angle:

I used to think that an object needed to have just the right velocity to enter orbit, but now we see that a wide range of velocities result in elliptical orbits.

Not surprisingly, ellipses are the basis of orbital mechanics for spaceflight. For example, imagine a spaceship in a circular orbit around the Earth. Firing its rockets briefly to speed up puts it into an elliptical orbit that takes it further away from the Earth on its opposite side and then returns it to its starting point. If it briefly fires its rockets at its furthest point, it can enter a circular orbit at a larger radius.