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Definition of acceleration for linear v−t graphs

Galileo’s experiment with dropping heavy and light objects from a tower showed that all falling objects have the same motion, and his inclined-plane experiments showed that the motion was described by v=at+vov=at+vo. The initial velocity depends on whether you drop the object from rest or throw it down, but even if you throw it down, you cannot change the slope of the v−t graph.

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Since these experiments show that all falling objects have linear v−t graphs with the same slope, the slope of such a graph is apparently an important and useful quantity. We use the word acceleration, and the symbol aa for the slope of such a graph. In symbols, a=Δv/Δa=Δv/Δt. The acceleration can be interpreted as the amount of speed gained in every second, and it has units of velocity divided by time, i.e., “meters per second per second,” or m/s/s. Continuing to treat units as if they were algebra symbols, we simplify “m/s/s” to read “m/s2“m/s2.” Acceleration can be a useful quantity for describing other types of motion besides falling, and the word and the symbol “aa” can be used in a more general context. We reserve the more specialized symbol “gg” for the acceleration of falling objects, which on the surface of our planet equals 9.8 m/s29.8 m/s2. Often when doing approximate calculations or merely illustrative numerical examples, it is good enough to use g=10 m/s2g=10 m/s2, which is off by only 2%.

The acceleration of gravity is different in different locations

Everyone knows gravity is weaker on the moon, but actually it is not even the same everywhere on Earth, as shown by the sampling of numerical data in the following table.

location latitude elevation (m) g textupmtextups2)
north pole 90∘N 9.8322
Reykjavik, Iceland 64∘N 0 9.8225
Guayaquil, Ecuador 2∘S 0 9.7806
Mt. Cotopaxi, Ecuador 1∘S 5896 9.7624
Mt. Everest 28∘N 8848 9.7643

The main variables that relate to the value of gg on Earth are latitude and elevation. Although you have not yet learned how gg would be calculated based on any deeper theory of gravity, it is not too hard to guess why gg depends on elevation. Gravity is an attraction between things that have mass, and the attraction gets weaker with increasing distance. As you ascend from the seaport of Guayaquil to the nearby top of Mt. Cotopaxi, you are distancing yourself from the mass of the planet. The dependence on latitude occurs because we are measuring the acceleration of gravity relative to Earth’s surface, but Earth’s rotation causes Earth’s surface to fall out from under you.

Much more spectacular differences in the strength of gravity can be observed away from Earth’s surface:

g textupmtextups2)

asteroid Vesta (surface)

Earth’s moon (surface)

Mars (surface)

Earth (surface)

Jupiter (cloud-tops)

Sun (visible surface)

typical neutron star (surface)

A typical neutron star is not so different in size from a large asteroid, but its orders of magnitude is more massive, so the mass of a body definitely correlates with the gg it creates. On the other hand, a neutron star has about the same mass as our sun, so why is its gg billions of times greater? If you had the misfortune of being on the surface of a neutron star, you would be within a few thousand miles of all its mass, whereas on the surface of the sun, you would still be millions of miles from most of its mass.

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