Acceleration
Discussion
definition
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.)
Units
International units (SI)
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 weightless at 0 g, normal at 1 g, twice as heavy at 2 g, and unconscious at 10 g. This unit has a precisely defined value of…
g = 9.80665 m/s2
but for everyday use 9.8 m/s2 is often 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.
- 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 coasters rarely exceed 30 m/s (100 kph, 60 mph). Contrary to popular belief, it is the acceleration that makes the ride interesting. A well designed roller coaster will subject the rider to maximum accelerations of 3 to 4 g for brief periods. This is what gives the ride its dangerous feel.
- Despite the immense power of its engines, the acceleration of the Space Shuttle was kept below 3 g. Anything greater would put unnecessary stress on the astronauts, the payload, and the ship itself. Once in orbit, the whole system enters into 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. (More on this later.)
- Fighter pilots in extreme situations 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 "g-force loss of consciousness" (G-LOC), fighter pilots wear special pressure suits that squeeze the legs and abdomen, forcing blood to remain in the head.
- Pilots and astronauts may also train in human centrifuges capable of up to 15 g. Exposure to such intense accelerations is kept 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.
Gaussian units (CGS)
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…
1 Gal = 1 cm/s2 = 0.01 m/s2
Did you catch that? The name of the unit is written all in lowercase (gal) while the symbol is capitalized (Gal). No other unit follows this strange pattern as far as I know. The gal was named in honor of the Italian physicist-astronomer-engineer Galileo Galilei (1564–1642) who was the first person to study the acceleration due to gravity in a systematic way, making him perhaps the first experimental scientist in history.
More popular than the gal in the world of gravimetry is the thousandth of a gal or milligal (mGal).
1 mGal = 0.001 Gal = 10−3 cm/s2 = 10−5 m/s2
The gal and milligal are a part of the Gaussian system of units, also known as the centimeter-gram-second (CGS) system. The Gaussian system may no longer be maintained as an active standard, but it gave us a handful of units that are in some sense "better" than their International System counterparts — better because they seem to be sized right. The milligal is possibly the "best" unit for gravimetry because the value of standard gravity stated in milligal has no digits to the right of the decimal point.
In SI units, the value of standard gravity…
1 g = 9.80665 m/s2
…has the classic, one non-zero digit to the left of the decimal point format. It looks respectable in a table with other similarly formatted numbers. The same quantity expressed in gal…
1 g = 980.665 Gal
…has a symmetry around the decimal point — 3 digits to the left, 3 to the right. That's nice, I guess. But the real winner is the milligal…
1 g = 980,665 mGal
…which makes standard gravity into a 6 digit whole number. Since sea level values of gravity on the surface of the Earth only vary in size by about one half of one percent, deviations in its value (Δg) expressed in milligal end up reasonably sized.
| gpolar | − | gstandard | = | Δg |
| 983,218 mGal | − | 980,665 mGal | = | +2,553 mGal |
| gequatorial | − | gstandard | = | Δg |
| 978,032 mGal | − | 980,665 mGal | = | −2,633 mGal |
| gsurface average | − | gstandard | = | Δg |
| 979,764 mGal | − | 980,665 mGal | = | −901 mGal |
| gsea level, 45° north | − | gstandard | = | Δg |
| 980,619 mGal | − | 980,665 mGal | = | −46 mGal |
| gcentral park, nyc | − | gstandard | = | Δg |
| 980,239 mGal | − | 980,665 mGal | = | −426 mGal |
If these calculations were done using m/s2 instead of mGal, the results would have absolute values less than 0.03.
gravitational anomaly need to be written
Somewhat acceptable non-SI units
kph/s
British-American units
ft/s2, mph/s
Here are some sample accelerations to end this section.
| 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 |
| 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 |
| 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 |