The Physics
Opus in profectus


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When the velocity of an object changes it is said to be accelerating. 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. For many Americans, their only experience with acceleration comes from car ads. When a commercial shouts "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 (called centripetal acceleration). Yes, that's right, a change in the direction of motion results in an acceleration even if the moving object neither sped up nor slowed down. That's because acceleration depends on the 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 the moon in orbit around the Earth accelerates. Acceleration occurs anytime an object's speed increases or decreases, or it changes direction.

Much like velocity, there are two kinds of acceleration: average and instantaneous. Average acceleration is determined over a "long" time interval. The word long in this context means finite — something with a beginning and an end. The velocity at the beginning of this interval is called the initial velocity, represented by the symbol v0 (vee nought), and the velocity at the end is called the final velocity, represented by the symbol v (vee). Average acceleration is a quantity calculated from two velocity measurements.

a =  v  =  v − v0
t t

In contrast, instantaneous acceleration is measured over a "short" time interval. The word short in this context means infinitely small or infinitesimal — having no duration or extent whatsoever. It's a mathematical ideal that can only be realized as a limit. The limit of a rate as the denominator approaches zero is called a derivative. Instantaneous acceleration is then the limit of average acceleration as the time interval approaches zero — or alternatively, acceleration is the derivative of velocity.

a = 
v  =  dv
t dt

Acceleration is the derivative of velocity with time, but velocity is itself the derivative of position with time. The derivative is a mathematical operation that can be applied multiple times to a pair of changing quantities. Doing it once gives you a first derivative. Doing it twice (the derivative of a derivative) gives you a second derivative. That makes acceleration the first derivative of velocity with time and the second derivative of position with time.

a =  dv  =  d   ds  =  d2s
dt dt dt dt2

A word about notation. In formal mathematical writing, vectors are written in boldface. Scalars and the magnitudes of vectors are written in italics. Numbers, measurements, and units are written in roman (not italic, not bold, not oblique — ordinary text). For example…

a = 9.8 m/s2, θ = −90°  or  a = 9.8 m/s2 at −90°

(Design note: I think Greek letters don't look good on the screen when italicized so I have decided to ignore this rule for Greek letters until good looking Greek fonts are the norm on the web.)


international units

Calculating acceleration involves dividing velocity by time — or in terms of SI units, dividing the meter per second [m/s] by the second [s]. Dividing distance by time twice is the same as dividing distance by the square of time. Thus the SI unit of acceleration is the meter per second squared.

m  =  m/s  =  m   1

s2 s s s

natural units

Another frequently used unit is the standard acceleration due to gravity — g. Since we are all familiar with the effects of gravity on ourselves and the objects around us it makes for a convenient standard for comparing accelerations. Everything feels normal at 1 g, twice as heavy at 2 g, and weightless at 0 g. This unit has a precisely defined value of 9.80665 m/s2, but for everyday use 9.8 m/s2 is sufficient, and 10 m/s2 is convenient for quick estimates.

The unit called the standard acceleration due to gravity (represented by a roman g) is not the same as the natural phenomenon called acceleration due to gravity (represented by an italic g). The former has a defined value whereas the latter has to be measured. (More on this later.)

Although the term "g force" is often used, the g is a measure of acceleration, not force. (More on forces later.) Of particular concern to humans are the physiological effects of acceleration. To put things in perspective, all values are stated in g.

Gaussian units

The precise measurement of the strength of gravity over the surface of the Earth or other celestial objects is called gravimetry. For historical reasons the preferred unit in this field is the centimeter per second squared also known as the gal. In symbolic form…

[Gal = cm/s2]

Yes, that's right. The name of the unit is written all in lowercase (gal) while the symbol is capitalized (Gal). The gal was named in honor of the Italian scientist Galileo Galilei (1564–1642) who was the first scientist to study the acceleration due to gravity — and maybe was the first scientist of any sort. Since the acceleration due to gravity varies by only small amounts over the surface of most celestial objects, deviations in strength from idealized models (called gravitational anomalies) are measured in thousandths of a gal or milligals (mGal).

[1,000 mGal = 1 Gal]

The gal and milligal are part of a precursor to the International System of Units called the centimeter-gram-second system or Gaussian system of units. I may one day actually write something significant in that section of this book.

Here are some sample accelerations to end this section.

Acceleration of selected events (smallest to largest)
a (m/s2) device, event, phenomenon, process
0 stationary or moving at a constant velocity
5 × 10−14 smallest acceleration in a scientific experiment
2.32 × 10−10 galactic acceleration at the Sun
9 × 10−10 anomalous acceleration of Pioneer spacecraft
0.5 elevator, hydraulic
0.63 free fall acceleration on Pluto
1 elevator, cable
1.6 free fall acceleration on the moon
8.8 International Space Station in orbit
3.7 free fall acceleration on Mars
9.8 free fall acceleration on Earth
10–40 manned rocket at launch
20 space shuttle, peak
24.8 free fall acceleration on Jupiter
29.06 Fastest 0–100 km/h by an electric car, average, 2023
37.36 Fastest 0–100 km/h by an electric car, peak, 2023
20–50 roller coaster
80 limit of sustained human tolerance
0–150 human training centrifuge
100–200 ejection seat
270 free fall acceleration on the Sun
600 airbags automatically deploy
104–106 medical centrifuge
~106 bullet in the barrel of a gun
~106 free fall acceleration on a white dwarf star
~1012 free fall acceleration on a neutron star
Automotive accelerations (g)
  starting braking cornering
typical car 0.3–0.5 0.8–1.0 0.7–0.9
sports car 0.5–0.9 1.0–1.3 0.9–1.0
F-1 race car 1.7 2 3
large truck <0.2 ~0.6  
Acceleration and the human body Primary source:Acceleration perturbations of daily living, 1994
a (g) event
02.9 sneeze
03.5 cough
03.6 crowd jostle
04.1 slap on back
08.1 hop off step
10.1 plop down in chair
60 chest acceleration 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