Acceleration
Cut to the Chase
When the velocity of an object changes it is said to be accelerating or more formally acceleration is the rate of change of velocity with time.
In everyday English, the word acceleration is often used to describe a state of increasing speed. When a car commercial quotes "zero to sixty in six point seven seconds" what they're saying here is that this particular car takes 6.7 s to reach a speed of 60 mph starting from a complete stop. This example illustrates acceleration as it is commonly understood, but acceleration in physics is much more than just increasing speed.
Any change in the velocity of an object results in an acceleration: increasing speed (what people usually mean when they say acceleration), decreasing speed (also called deceleration or retardation), or changing direction. Yes, that's right, a change in the direction of motion results in an acceleration even if the speed didn't change. That's because acceleration depends on a change in velocity and velocity is a vector quantity  one with both magnitude and direction. Thus, a falling apple accelerates, a car stopping at a traffic light accelerates, and an orbiting planet accelerates. Acceleration occurs anytime an object's speed increases, speed decreases, or direction of motion changes.
Average Acceleration 

Instantaneous Acceleration 

As was the case with velocity, there are two kinds of acceleration  average and instantaneous. Average acceleration is measured over a "long" (measurable) time interval while instantaneous acceleration is measured over a "very small" (infinitesimal) time interval. The formulas are also similar to those of velocity. Acceleration is defined by the following equations …
For those of you familiar with calculus, check out the second equation, which states that acceleration is the first derivative of velocity with respect to time and the second derivative of displacement with respect to time. Or if you prefer, acceleration is the rate of change in velocity and also (since velocity is a change in displacement) the rate of change of the rate of change of displacement.
units
Calculating acceleration involves dividing velocity by time  or in terms of units, dividing meters per second [m/s] by second [s]. Dividing distance by time twice is the same as dividing distance by the square of time. The SI unit of acceleration is the meter per second squared [m/s^{2}].
Another frequently used unit is the acceleration due to gravity. Since we are all familiar with the effects of gravity on ourselves and the objects around us it makes for a convenient standard against which we can compare other accelerations. Everything feels normal at 1 g, twice as heavy at 2 g, and weightless at 0 g. This unit has a very precise definition, but for everyday use 9.8 m/s^{2} is sufficient.
The unit called the acceleration due to gravity, with the symbol g (roman), is not the same as the natural phenomena (discussed in a later section) called the acceleration due to gravity, having the symbol g (italic). The former is a constant whereas the latter is a variable.
Although the term "g force" is often used, the g is a measure of acceleration and not force. Forces are discussed at length in the next chapter.
The acceleration due to gravity …
is a natural unit of acceleration,
is represented by the symbol g (roman),
is equal to 9.80665 m/s^{2} by definition,
is often rounded to 9.8 m/s^{2} for convenience, and
is sometimes called the "g force" even though it is not a measure of force.
Of particular concern to humans are the physiological effects of acceleration. To put things in perspective, all values are stated in g.
In roller coaster design, speed is of the essence. Or, is it? If speed was all there was to designing a thrill ride, then the freeway would be pretty exciting. Most roller coaster rarely exceed 30 m/s (60 mph). Contrary to popular belief, it is the acceleration that makes the ride. A well designed roller coaster will subject the rider to maximum accelerations of 3 to 4 g for brief periods. This gives the ride a faster, more dangerous feel.
Despite the immense power of its engines, the acceleration of the Space Shuttle is kept below 3 g. Anything greater would put unnecessary stress on the astronauts and the ship itself. Once in orbit, they enter an extended period of free fall, which provides the sensation of weightlessness. Such a "zero g" environment can also be simulated inside a specially piloted aircraft or a free fall drop tower.
Fighter pilots can experience accelerations of up to 8 g for brief periods during tactical maneuvers. If sustained for more than a few seconds, 4 to 6 g is sufficient to induce blackout. To prevent loss of consciousness, fighter pilots wear special pressure suits that squeeze the legs and abdomen, forcing blood to remain in the head.
Pilots and astronauts often train in human centrifuges capable of up to 15 g. Exposure to such intense accelerations is kept very brief for safety reasons. Humans are rarely subjected to anything higher than 8 g for longer than a few seconds.
Acceleration is related to injury. This is why the most common sensor in a crash test dummy is the accelerometer. Extreme acceleration can lead to death. The acceleration during the crash that killed Diana, Princess of Wales, in 1997 was estimated to have been on the order of 70 to 100 g, which was intense enough to tear the pulmonary artery from her heart  an injury that is nearly impossible to survive. Had she been wearing a seat belt, the acceleration would have been something more like 30 or 35 g  enough to break a rib or two, but not nearly enough to kill most people.
Automotive Acceleration (g)  
event  typical car  sports car  F1 race car  large truck 

starting  0.3  0.5  > 0.9  1.7  < 0.2 
braking  0.8  1.0  > 1.3  2  ~ 0.6 
cornering  0.6  1.0  > 2.5  3 
Acceleration and the Human Body  
a (g)  event 

2.9  sneeze 
3.5  cough 
3.6  crowd jostle 
4.1  slap on back 
8.1  hop off step 
10.1  plop down in chair 
60  chest acceleration limit during car crash at 48 km/h with airbag 
70  100  crash that killed Diana, Princess of Wales, 1997 
150  200  head acceleration limit during bicycle crash with helmet 
Source: Spine, June 1994 
Summary
Acceleration is the rate of change of velocity with time.
As a vector it must be stated with both magnitude and direction.
Acceleration occurs anytime an object's …
speed increases,
speed decreases, or
direction of motion changes.
The symbol for acceleration is a (boldface). The SI unit of acceleration is the meter per second squared [m/s^{2}].
The acceleration due to gravity …
is a natural unit of acceleration,
is represented by the symbol g (roman),
is equal to 9.80665 m/s^{2} by definition,
is often rounded to 9.8 m/s^{2} for convenience, and
is sometimes called the "g force" even though it is not a measure of force.
Average acceleration is measured over a nonzero time interval.
Instantaneous acceleration is the limit of average acceleration as the time interval approaches zero.
In the language of calculus, instantaneous acceleration is …
the first derivative of velocity with respect to time and
the second derivative of displacement with respect to time.
Acceleration is defined by the following equations …
average acceleration  a =  Δv  
Δt  
instantaneous acceleration  a =  lim  Δv  =  dv  =  d^{2}r  
Δt→0  Δt  dt  dt^{2}  