The first mathematical formulation of gravity was published in 1687 by Sir Isaac Newton. His law of universal gravitation was the standard theory of gravity until work by Albert Einstein and others on general relativity. Since calculations in general relativity are complicated, and Newtonian gravity is sufficiently accurate for most applications, when dealing with weak gravitational fields (e.g., when launching rockets, projectiles, or swinging pendulums, etc.), Newton's formulas are generally preferred.
Steven Weinberg noted that we could still build a suspension bridge with Newton's laws. Newton's theory is much simpler in mathematical structure than general relativity, and is used very often. Newton's Laws are generally taught at the high school level while general relativity is taught only to students opting for physics in undergraduate and graduate courses.
Stars in the gravitational field of a globular cluster, M80.
Newton conceived of gravitation when he considered the trajectory of a projectile . A projectile under the influence of gravity travels along a trajectory that is a conic section. The projectile follows either an elliptical path or a hyperbolic path, depending on whether its total mechanical energy is less than or greater than that necessary for escape velocity, respectively. In the pathological case where the projectile's total mechanic energy is exactly equal to that necessary for escape velocity, the projectile follows a parabolic trajectory. At low speeds and over small distances (small enough that the surface of the Earth can be considered flat), the elliptical trajectory of a projectile can be more easily approximated as a parabolic trajectory.
When Newton heard the sound of an apple falling on the ground, he asked himself might the same cause (which he called gravitation) for the motion of the apple, also explain the motion of the moon?. This was the first statement of the universal law of gravitation.
Einstein's theory of general relativity (1915) stated that the presence of mass, energy, and momentum causes spacetime to become curved. Because of this curvature, the paths that objects in inertial motion follow can "deviate" or change direction over time. This deviation appears to us as an acceleration towards massive objects, which Newton characterized as being gravity. In general relativity however, this acceleration or free fall is actually inertial motion. So objects in a gravitational field appear to fall at the same rate due to their being in inertial motion while the observer is the one being accelerated. (This identification of free fall and inertia is known as the Equivalence principle.)
Astronaut in motion above Earth's horizon. The atmosphere can be glimpsed as a fuzzy layer above the enormous mass of Earth. Space, a region with far less air, is the dark background behind the astronaut and the earth.
Galileo was the first to demonstrate and then formulate the equation for the distance d traveled by a falling object under the influence of gravity g for a time t:
He used a woodmolding, "12 cubits long, half a cubit wide and three finger-breadths thick" as a ramp with a straight, smooth, polished groove to study rolling balls ("a hard, smooth and very round bronze ball"). He lined the groove with "parchment, also smooth and polished as possible". He inclined the ramp at various angles, effectively slowing down the acceleration enough so that he could measure the elapsed time. He would let the ball roll a known distance down the ramp, and used a water clock to measure the time taken to move the known distance; this clock was
"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.".
We can distinguish several meanings for mass in physics:
Inertial mass is a measure of an object's inertia: its resistance to changing its state of motion when a force is applied. An object with small inertial mass changes its motion more readily, and an object with large inertial mass does so less readily. Now we know that this mass is actually a measure of the total energy of the object and is equal , where E is total energy of the object and c is speed of light in a vacuum. Not only inertial but also all gravitational effects regarding this object are proportional to the object's inertial mass and that's why it is exactly the same as "gravitational mass", "passive" and "active". Before discovery of Einstein's theory of gravitation the "gravitational masses" were considered separate physical entities.
Passive gravitational mass is a measure of the strength of an object's interaction with the gravitational field. Within the same gravitational field, an object with a smaller passive gravitational mass experiences a smaller gravitational force than an object with a larger passive gravitational mass. This force is called the weight of the object. In informal usage, the word "weight" is often used synonymously with "mass", because the strength of the gravitational field is roughly constant everywhere on the surface of the Earth. In physics, the two terms are distinct: an object will have a larger weight if it is placed in a stronger gravitational field, but its passive "gravitational mass" (inertial mass) remains unchanged.
Active gravitational mass is a measure of the strength of the gravitational field due to a particular object. For example, the gravitational field that one experiences on the Moon is weaker than that of the Earth because the Moon has less "active gravitational mass" than the Earth.
Note that Einstein wrote, in his paper on General Relativity that inertial mass and gravitational mass must be equivalent. "We see that our extension of the principle of relativity implies the necessity of the law of the equality of inertial and gravitational mass."-Einstein (1916)