Acceleration

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 inclinedplane experiments showed that the motion was described by v = at + v_o. The initial velocity v_o 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, a, of the v - t graph.

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 a, for the slope of such a graph. In symbols, a = 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." Acceleration can be a useful quantity for describing other types of motion besides falling, and the word and the symbol "a" can be used in a more general context. We reserve the more specialized symbol "g" for the acceleration of falling objects, which on the surface of our planet equals 9.8 m/s². Often when doing approximate calculations or merely illustrative numerical examples it is good enough to use g = 10 m/s², which is off by only 2%.

Finding final speed, given time.

Extracting acceleration from a graph.

Converting g to different units.

The acceleration of gravity is different in different locations.

Everyone knows that 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 (m/s2)
north pole	90°N	0	9.8322
Reykjavik, Iceland	64°N	0	9.8225
Fullerton, California	34°N	0	9.7957
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 g on Earth are latitude and elevation. Although you have not yet learned how g would be calculated based on any deeper theory of gravity, it is not too hard to guess why g 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 the earth's surface, but the earth's rotation causes the earth's surface to fall out from under you. (We will discuss both gravity and rotation in more detail later in the course.)

h / This false-color map shows variations in the strength of the earth's gravity. Purple areas have the strongest gravity, yellow the weakest. The overall trend toward weaker gravity at the equator and stronger gravity at the poles has been artificially removed to allow the weaker local variations to show up. The map covers only the oceans because of the technique used to make it: satellites look for bulges and depressions in the surface of the ocean. A very slight bulge will occur over an undersea mountain, for instance, because the mountain's gravitational attraction pulls water toward it. The US government originally began collecting data like these for military use, to correct for the deviations in the paths of missiles. The data have recently been released for scientific and commercial use (e.g., searching for sites for off-shore oil wells).

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

location	g (m/s2)
asteroid Vesta (surface)	0.3
Earth's moon (surface)	1.6
Mars (surface)	3.7
Earth (surface)	9.8
Jupiter (cloud-tops)	26
Sun (visible surface)	270
typical neutron star (surface)	1012
black hole (center)	infinite according to some theories, on the order of 1052 according to others

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

Discussion Questions

A	What is wrong with the following definitions of g? (1) "g is gravity." (2) "g is the speed of a falling object." (3) "g is how hard gravity pulls on things."
B	When advertisers specify how much acceleration a car is capable of, they do not give an acceleration as defined in physics. Instead, they usually specify how many seconds are required for the car to go from rest to 60 miles/hour. Suppose we use the notation "a" for the acceleration as defined in physics, and "acar ad" for the quantity used in advertisements for cars. In the US's non-metric system of units, what would be the units of a and acar ad? How would the use and interpretation of large and small, positive and negative values be different for a as opposed to acar ad?
C	Two people stand on the edge of a cliff. As they lean over the edge, one person throws a rock down, while the other throws one straight up with an exactly opposite initial velocity. Compare the speeds of the rocks on impact at the bottom of the cliff.