3.2    Free-Fall, Orbit, and Weightlessness

If a ball is thrown near the surface of the Earth, it arcs toward the Earth.  At the human scale of size, force, and speed, one can reasonably assume that the Earth is motionless and flat and that the lines of force of gravity are parallel.  Then the instantaneous velocity of the ball consists of two components: a constant horizontal velocity (neglecting air resistance) established by the initial throw, and a vertical velocity that increases in magnitude under the influence of gravity toward the surface of the Earth.  With this flat Earth assumption, it is easily shown that the ball follows a parabolic trajectory:

[math]

(3.5)

where:

 X_dot_dot, Z_dot_dot  are the horizontal and vertical accelerations.

 X_dot, Z_dot  are the horizontal and vertical velocities at time t.

 X, Z  are the horizontal and vertical positions at time t.

 X_dot_0, Z_dot_0, X0, Z0  are the initial velocities and positions at time 0.

Holding the other initial conditions constant, increasing the horizontal velocity X_dot_0 increases the span of the arc.  If the ball is thrown very hard, it may travel so far that the flat Earth assumption is no longer valid: the curvature of the Earth's surface within the span of the arc becomes significant.  If one now assumes that the Earth is a homogeneous sphere with the lines of force of gravity radiating toward its center, and continues to ignore air resistance and the gravitational attraction of all other bodies, then describing the ball's motion becomes a central force problem.  By Kepler's first law, its motion is described by a conic section with one focus at the center of the Earth.  (Pollard [6] gives a proof.)  The eccentricity of the conic depends on the energy imparted to the ball:  if it is moving at less than escape velocity, its path is elliptical; if exactly escape velocity, parabolic; if greater than escape velocity, hyperbolic.  With the spherical Earth assumption, the parabola described by equation 3.5 is only an approximation of the real trajectory, which is an extremely long narrow ellipse.  Every ball ever thrown would orbit the center of the Earth in such an ellipse if the bulk of the Earth itself didn't get in the way.  If the ball could be thrown hard enough, the ellipse would become sufficiently large to miss the Earth entirely, and the ball would in fact orbit the Earth: it would become a satellite.

Figure 3.3

Figure 3.3:  Free-fall, orbit, and weightlessness.

The question, then, is not "What holds a satellite up?" but rather "What holds it down?"  According to the first law of motion, the Moon, the space station, and all other satellites would fly off in straight lines to infinity if gravity didn't act to hold them down.  At the 350 kilometer altitude planned for the space station, Earth's gravity continues to pull with 90 percent of the intensity we feel at its surface.  Nothing holds a space station up; the station and the astronauts within it fall continuously toward the Earth, but because of their high velocity they continuously overshoot and miss the Earth entirely.

Weightlessness, then, is simply free-fall.  Astronauts are not weightless relative to the Earth, but only relative to their immediate surroundings, which offer no platform upon which to bear their weight.  Keeping this in mind helps one to appreciate the problem of "space adaptation syndrome":  think of a diving board, a high ladder, or a tall building, and imagine falling, for days or weeks or months at a time.