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Metre per second squared
Unit systemSI
Unit ofacceleration
Symbol⁠m/s2⁠

The metre per second squared or metre per square second is the unit of acceleration in the International System of Units (SI). As a derived unit, it is composed from the SI base units of length, the metre, and of time, the second. Its symbol is written in several forms as m/s2, m·s−2 or ms−2, , or less commonly, as (m/s)/s.[1]

As acceleration, the unit is interpreted physically as change in velocity or speed per time interval, i.e. metre per second per second and is treated as a vector quantity.

Example

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When an object experiences a constant acceleration of one metre per second squared (1 m/s2) from a state of rest, it achieves the speed of 5 m/s after 5 seconds and 10 m/s after 10 seconds. The average acceleration a can be calculated by dividing the speed v (m/s) by the time t (s), so the average acceleration in the first example would be calculated:

.
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Newton's second law states that force equals mass multiplied by acceleration. The unit of force is the newton (N), and mass has the SI unit kilogram (kg). One newton equals one kilogram metre per second squared. Therefore, the unit metre per second squared is equivalent to newton per kilogram, N·kg−1, or N/kg.[2]

Thus, the Earth's gravitational field (near ground level) can be quoted as 9.8 metres per second squared, or the equivalent 9.8 N/kg.

Acceleration can be measured in ratios to gravity, such as g-force, and peak ground acceleration in earthquakes.

Unicode character

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The "metre per second squared" symbol is encoded by Unicode at code point U+33A8 ㎨ SQUARE M OVER S SQUARED. This is for compatibility with East Asian encodings and not intended to be used in new documents.[3]

Conversions

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Conversions between common units of acceleration
Base value (Gal, or cm/s2) (ft/s2) (m/s2) (Standard gravity, g0)
1 Gal, or cm/s2 1 0.0328084 0.01 1.01972×10−3
1 ft/s2 30.4800 1 0.304800 0.0310810
1 m/s2 100 3.28084 1 0.101972
1 g0 980.665 32.1740 9.80665 1

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Metre per second squared

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from Grokipedia
The metre per second squared (symbol: m/s² or m⋅s⁻²) is the derived unit of acceleration in the International System of Units (SI), expressing the rate at which an object's velocity changes over time—specifically, an increase of one metre per second in velocity for each second elapsed.[1] This unit is formed from the SI base units of length (the metre, m) and time (the second, s), with the metre defined as the distance light travels in vacuum in 1/299,792,458 of a second and the second defined as the duration of 9,192,631,770 periods of radiation corresponding to the transition between two hyperfine levels of the caesium-133 ground state. In physics, m/s² quantifies acceleration in contexts such as Newton's second law (F = m⋅a), where force equals mass times acceleration, and it underpins derived units like the newton (N = kg⋅m⋅s⁻²) for force. A key reference value is the standard acceleration due to gravity (g_n), defined exactly as 9.80665 m/s², which approximates Earth's gravitational pull at sea level and serves as a benchmark in engineering and metrology.[2] Since the 2019 SI redefinition, m/s² benefits from enhanced stability tied to invariant physical constants, ensuring precise, reproducible measurements across scientific disciplines.[1]

Definition and Fundamentals

Core Definition

The metre per second squared, symbol m/s², is the coherent derived SI unit of acceleration, defined as the rate of change of velocity per unit time, specifically one metre per second divided by one second. This unit quantifies acceleration, which is a vector quantity describing how velocity changes over time. The dimensional formula for the metre per second squared is [L][T]^{-2}, where [L] denotes length and [T] denotes time.[3][4] It is derived directly from the SI base units of the metre (m) for length and the second (s) for time, expressed as m · s^{-2}, requiring no special physical artifact or additional constant for its realization beyond the definitions of the base units themselves.[3][4] As a coherent SI unit, the metre per second squared incorporates no numerical factor other than unity when combined with other SI base or derived units in physical equations, ensuring dimensional consistency throughout the system.[3][4] This coherence facilitates precise calculations in mechanics and related fields without the need for conversion coefficients.[3]

Relation to Acceleration

The metre per second squared (m/s²) serves as the SI derived unit for measuring the magnitude of acceleration, a vector quantity that describes the time rate of change of velocity.[5] In physics, acceleration quantifies how quickly an object's velocity vector—encompassing both speed and direction—alters over time, with its magnitude expressed in m/s² to reflect the change in velocity (in metres per second) per second.[6] For motion under constant acceleration, this relationship is formalized by the equation
a=ΔvΔt, a = \frac{\Delta v}{\Delta t},
where aa is the acceleration in m/s², Δv\Delta v is the change in velocity, and Δt\Delta t is the time interval.[7] This definition underscores acceleration's role in describing deviations from uniform motion, distinguishing it from velocity (measured in m/s), which tracks displacement over time, and from jerk (measured in m/s³), the rate of change of acceleration itself.[8] In Newtonian mechanics, acceleration bridges the description of linear motion—where velocity changes only in magnitude—to curved paths, where changes in direction also contribute to the overall vector alteration.[9] Under uniform acceleration, the displacement of an object follows a quadratic dependence on time, given by the kinematic equation
s=ut+12at2, s = ut + \frac{1}{2}at^2,
where ss is the displacement, uu is the initial velocity, tt is the time elapsed, and aa is the constant acceleration in m/s².[10] This equation arises from integrating the constant acceleration over time, highlighting how sustained acceleration produces non-linear position changes, essential for analyzing varied motion profiles in classical kinematics. Acceleration in m/s² also features prominently in Newton's second law of motion, which states that the net force FF acting on an object equals its mass mm times its acceleration aa:
F=ma. F = ma.
Here, force is quantified in newtons (N), where 1 N is defined as the force required to accelerate a 1 kg mass at 1 m/s², yielding the dimensional equivalence N = kg⋅m/s².[11] This law establishes acceleration as the direct response to unbalanced forces, linking kinematics to dynamics and enabling the prediction of motion from applied influences.[12]

Notation and Standards

Symbolic Representation

The standard symbol for the metre per second squared, the SI derived unit of acceleration, is m/s², where the solidus (/) denotes division by the square of the second.[3] This notation follows the rules for expressing derived units in the [International System of Units](/page/International_System_of Units) (SI), ensuring clarity in scientific expressions.[3] Alternatively, for greater precision in complex formulas, the unit may be written as m⋅s⁻², using a multiplication dot (⋅) and a negative exponent to indicate the inverse square of the second.[3] This form avoids ambiguity when combining with other units, as recommended in SI guidelines.[13] Typography for the symbol requires the superscript ² to be properly raised, rendered in upright (roman) font without italics, and with no spaces around the solidus (e.g., 9.8 m/s² rather than 9.8 m/s2 or m/s²).[3] In LaTeX typesetting, the basic form is achieved with \mathrm{m/s^2} to ensure roman font and correct superscript positioning, while the siunitx package provides \si{\metre\per\second\squared} for automated, standards-compliant rendering including proper spacing and localization.[14] These conventions promote consistency across scientific literature and digital documents.[13] The full name is "metre per second squared" in singular form, pluralized as "metres per second squared" when referring to multiple instances, though the symbol m/s² remains unchanged regardless of quantity.[3] Unit symbols in the SI do not inflect for plurality, maintaining uniformity in technical writing.[3] The symbol m/s² is consistent across international variants of the SI, including English and French, where the name translates to "mètre par seconde carrée" but retains the identical symbol.[15] This standardization facilitates global scientific communication without altering notation based on language.[3]

Metrological Standards

Following the 2019 redefinition of the International System of Units (SI), the metre per second squared (m/s²) is realized as a derived unit through the fixed numerical values of fundamental constants, eliminating dependence on physical artifacts for its base components. The metre is defined by fixing the speed of light in vacuum to exactly 299 792 458 m/s, while the second is defined by fixing the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, Δν_Cs, to exactly 9 192 631 770 Hz. This ensures that acceleration, expressed as m/s², maintains absolute traceability to invariant properties of nature, with the unit itself serving as the coherent SI measure without need for prototypes or reference objects.[16][3] Practical realization of the m/s² unit in laboratories involves combining high-precision measurements of length and time to quantify acceleration, typically through techniques such as laser interferometry for displacement and atomic clocks for timing. For instance, accelerometers are calibrated by comparing their output to motion generated in controlled setups, like vibration exciters or rotating arms, where acceleration is computed from twice-differentiated position data traceable to the speed of light and caesium frequency. These methods allow direct linkage to SI base units, often using fringe-counting interferometers to achieve sub-micrometre resolution in length over millisecond timescales. Calibration against gravitational standards, such as local free-fall measurements, provides additional verification while remaining anchored to the redefined constants.[17][18][19] The International Bureau of Weights and Measures (BIPM) plays a central role in upholding the m/s² unit by maintaining the official SI Brochure, which details its status as the base-form derived unit for acceleration and outlines guidelines for its use without prefixes in core definitions (though practical scales like mm/s² are permitted for small accelerations). The BIPM coordinates global metrology through consultative committees, ensuring consistency in realizations across national institutes and updating protocols to reflect advancements in atomic and optical standards. This framework supports worldwide uniformity in acceleration measurements for scientific and industrial applications.[3][1] In laboratory settings, typical standards for realizing m/s² achieve relative measurement uncertainties on the order of 0.1% (10^{-3}) or better, reflecting the precision of interferometric and chronometric tools in controlled environments. These uncertainties stem from minimized systematic errors in primary calibrations, enabling reliable dissemination of the unit to secondary standards with propagated confidence levels typically at the 0.1% level or better.[19][20][21]

Physical Contexts and Applications

Gravitational Acceleration

The metre per second squared serves as the SI unit for quantifying gravitational acceleration on Earth, where the standard value, denoted as $ g $, is defined exactly as 9.80665 m/s² to represent the nominal acceleration due to gravity at sea level and 45° latitude for metrological purposes.[2] This defined value facilitates consistent standardization across physical measurements and engineering applications.[22] The actual magnitude of gravitational acceleration varies geographically due to factors such as Earth's oblateness, which increases the distance from the center at the equator, and the centrifugal effect from rotation, which reduces the effective acceleration most noticeably at lower latitudes. An approximate formula capturing this latitudinal dependence is
g(ϕ)≈9.7803(1+0.0053sin⁡2ϕ−0.0000059sin⁡2(2ϕ)) m/s2, g(\phi) \approx 9.7803 \left(1 + 0.0053 \sin^2 \phi - 0.0000059 \sin^2 (2\phi)\right) \ \text{m/s}^2,
where $ \phi $ is the geodetic latitude in degrees; this yields values ranging from about 9.78 m/s² at the equator to 9.83 m/s² at the poles.[23] Additionally, $ g $ decreases with increasing altitude above sea level because of the inverse-square law of gravitation, with an example value of approximately 9.78 m/s² at 2000 m elevation near the equator, reflecting a small reduction of roughly 0.006 m/s² from sea-level conditions.[24] Measurements of local gravitational acceleration are conducted using precise instruments like gravimeters, which detect minute variations in $ g $ through changes in the position of a test mass, or simple pendulums, where the period $ T = 2\pi \sqrt{l/g} $ allows computation of $ g $ from known length $ l $ and observed oscillation time.[25] Free-fall experiments further verify $ g $ by dropping an object over a known distance $ s $ and timing the fall $ t $, applying the kinematic equation
s=12gt2 s = \frac{1}{2} g t^2
to solve for $ g $, often using laser interferometry for high accuracy in controlled settings.[26] Beyond Earth, the metre per second squared provides a universal measure for comparing planetary surface gravities, such as the Moon's approximate value of 1.62 m/s²—about 16% of Earth's[27]—or Mars's 3.71 m/s², roughly 38% of Earth's, enabling assessments of environmental impacts on exploration and biology across solar system bodies.[28]

Kinematics and Dynamics

In kinematics, the metre per second squared serves as the unit for acceleration in two-dimensional motion scenarios, such as projectile motion, where an object is launched with an initial velocity and subjected to constant acceleration due to gravity. The horizontal component of acceleration remains zero (ax=0a_x = 0), while the vertical component is directed downward (ay=−ga_y = -g), with gg denoting the gravitational acceleration in m/s²; this results in a parabolic trajectory for the object's path. In dynamics, the unit applies to centripetal acceleration in uniform circular motion, given by the formula ac=v2ra_c = \frac{v^2}{r}, where vv is the tangential speed in m/s and rr is the radius of the path in m, yielding acceleration toward the center with magnitude in m/s². This concept is essential in engineering applications, such as vehicle dynamics, where a car navigating a curve at typical street speeds might experience a lateral centripetal acceleration of around 5 m/s², requiring sufficient tire friction to maintain stability.[29] In special relativity, the metre per second squared measures proper acceleration, defined as the acceleration felt by an observer in their instantaneous rest frame, invariant under Lorentz transformations and contrasting with coordinate acceleration. While everyday applications remain in the Newtonian regime, relativistic effects become prominent in extreme scenarios, such as near black holes, where proper acceleration required to hover at fixed radial distance can reach enormously high values on the order of 101310^{13} m/s² for a stellar-mass black hole, far exceeding terrestrial scales.[30][31] Engineering applications, particularly in crash testing, consider human tolerance to acceleration measured in m/s² to design protective structures. For short durations (under 0.2 s), humans can withstand approximately 10–15 g (98–147 m/s²) along certain axes, such as forward-facing impacts, without severe injury, provided proper restraint; this threshold informs vehicle safety standards and impact simulations.[32]

Conversions and Equivalents

To Other Acceleration Units

The metre per second squared (m/s²) relates to other acceleration units through conversion factors derived from the definitions of their base length and time units, which are tied to the SI metre and second. In the imperial (foot-pound-second) system, acceleration is commonly expressed in feet per second squared (ft/s²). The exact relation stems from the definition 1 ft = 0.3048 m, so the acceleration scales inversely for the length unit:
1 m/s2=10.3048 ft/s2≈3.280839895 ft/s2. 1 \, \mathrm{m/s^2} = \frac{1}{0.3048} \, \mathrm{ft/s^2} \approx 3.280839895 \, \mathrm{ft/s^2}.
This factor is used in engineering and physics applications requiring imperial units.[33] The galileo (Gal), a unit from the centimetre-gram-second (CGS) system defined as exactly 1 cm/s², provides another bridge to imperial units. Since 1 m = 100 cm,
1 m/s2=100 cm/s2=100 Gal. 1 \, \mathrm{m/s^2} = 100 \, \mathrm{cm/s^2} = 100 \, \mathrm{Gal}.
Relating to ft/s², 1 ft = 30.48 cm exactly, so 1 ft/s² = 30.48 Gal, and thus
1 m/s2=10030.48 ft/s2≈3.28084 ft/s2, 1 \, \mathrm{m/s^2} = \frac{100}{30.48} \, \mathrm{ft/s^2} \approx 3.28084 \, \mathrm{ft/s^2},
confirming the imperial conversion via CGS intermediaries.[33] For expressions in multiples of the standard acceleration due to gravity $ g $, defined exactly as $ g = 9.80665 , \mathrm{m/s^2} $ for metrological purposes, the relation is
1 m/s2=19.80665 g≈0.101971621 g. 1 \, \mathrm{m/s^2} = \frac{1}{9.80665} \, g \approx 0.101971621 \, g.
This equivalence facilitates comparisons in fields like ballistics and vehicle dynamics.[2] In the full CGS system, acceleration is in cm/s² (identical to Gal), so the direct conversion remains
1 m/s2=100 cm/s2. 1 \, \mathrm{m/s^2} = 100 \, \mathrm{cm/s^2}.
This scaling arises because the metre-to-centimetre factor of 100 applies to velocity (m/s to cm/s), and thus again to acceleration.[33] In aviation and nautical contexts, acceleration uses knots per second (kn/s), where the knot is a non-SI speed unit accepted for use with the SI and defined via the international nautical mile of exactly 1852 m per hour (3600 s). Thus, 1 kn = 1852/3600 m/s exactly = 0.514444444 m/s, and
1 kn/s=0.514444444 m/s2, 1 \, \mathrm{kn/s} = 0.514444444 \, \mathrm{m/s^2},
so
1 m/s2=10.514444444 kn/s≈1.943844 kn/s. 1 \, \mathrm{m/s^2} = \frac{1}{0.514444444} \, \mathrm{kn/s} \approx 1.943844 \, \mathrm{kn/s}.
This unit appears in flight performance analyses and maritime engineering.[34] In some contexts, particularly road vehicle performance or long-duration motion, acceleration may be expressed in kilometres per hour squared (km/h²). The exact conversion is
1 m/s2=12 960 km/h2. 1 \, \mathrm{m/s^2} = 12\,960 \, \mathrm{km/h^2}.
This factor derives from 1 m/s = 3.6 km/h exactly (since 3600 s/h ÷ 1000 m/km = 3.6) and scaling by the seconds in an hour: 3.6 × 3600 = 12,960.[35]
UnitSymbolConversion from m/s²Notes
Foot per second squaredft/s²≈ 3.28084 ft/s²Derived from 1 ft = 0.3048 m exactly
GalileoGal= 100 GalCGS unit; 1 Gal = 1 cm/s²
Standard gravityg≈ 0.10197 gg = 9.80665 m/s² exactly by definition
Centimetre per second squaredcm/s²= 100 cm/s²Direct CGS base unit
Knot per secondkn/s≈ 1.94384 kn/s1 kn = 1852/3600 m/s exactly
Kilometre per hour squaredkm/h²= 12,960 km/h²Derived from 1 m/s = 3.6 km/h exactly and 1 h = 3600 s

Numerical Examples

To illustrate the use of metre per second squared (m/s²) in unit conversions, consider the standard gravitational acceleration, often approximated in imperial units as 32.2 ft/s². This value converts to m/s² by multiplying by the length conversion factor of 0.3048 m/ft, yielding approximately 32.2 × 0.3048 = 9.81 m/s², which aligns closely with the defined standard gravity of 9.80665 m/s².[36][37][2] In a practical kinematic scenario, a car accelerating from rest to 100 km/h in 10 seconds provides another example. First, convert 100 km/h to metres per second: 100 × (1000 m/km) / (3600 s/h) = 100 / 3.6 ≈ 27.78 m/s. The average acceleration is then a=ΔvΔt=27.78 m/s10 s=2.778 m/s2a = \frac{\Delta v}{\Delta t} = \frac{27.78 \, \mathrm{m/s}}{10 \, \mathrm{s}} = 2.778 \, \mathrm{m/s^2}.[38] For multi-unit comparisons, an elevator accelerating upward at 1.5 m/s² can be expressed relative to standard gravity and imperial units. Dividing by the standard value gives 1.5/9.80665≈0.153 g1.5 / 9.80665 \approx 0.153 \, g, where gg denotes standard gravity. To convert to feet per second squared, multiply by the factor 3.28084 ft/m: 1.5×3.28084≈4.921 ft/s21.5 \times 3.28084 \approx 4.921 \, \mathrm{ft/s^2}.[2][37] An additional example demonstrates conversion to km/h²: an acceleration of 4.37 m/s² equals 56,635.2 km/h², calculated as 4.37 × 12,960. Rounding in such conversions introduces minor errors; for instance, approximating 1 m/s² as 3.28 ft/s² yields a relative error of about 0.002% compared to the exact factor of 3.280839895 ft/s² derived from 1 ft = 0.3048 m.[37]

Historical Development

Origins in the Metric System

The metre per second squared (m/s²) emerged as a derived unit within the foundational framework of the metric system, which originated in France during the late 18th century amid the French Revolution. In 1791, the French Academy of Sciences proposed a decimal-based system of measurement to replace inconsistent local units, defining the metre as one ten-millionth of the distance from the Earth's equator to the North Pole along a meridian passing through Paris—a length determined through expeditions measuring the meridian arc from Dunkirk to Barcelona. This definition was provisionally adopted in 1795, with a provisional platinum metre bar created in 1793, and officially sanctioned in 1799 following the construction of a definitive platinum artifact standard. The second, as the unit of time, was retained from pre-metric traditions and defined as 1/86,400 of the mean solar day, providing a stable temporal base that allowed for the coherent derivation of acceleration as change in velocity per unit time, yielding m/s².[39][40] Pre-metric influences laid conceptual groundwork for such units, notably in the work of Galileo Galilei in the early 17th century. Galileo's experiments on falling bodies, conducted using inclined planes to slow motion for measurement, employed distances in cubits (approximately 0.45 meters) and time intervals approximated by seconds via pulse beats or water clocks, revealing that objects accelerate uniformly under gravity—distance fallen proportional to the square of time elapsed. This empirical insight into acceleration as a rate of velocity change implicitly aligned with the later m/s² formulation, though Galileo lacked a standardized length unit like the metre.[41][42] In the 19th century, as the metric system gained traction among scientists, physicists began formalizing acceleration in mechanics using metre and second equivalents. Gaspard-Gustave de Coriolis, in his 1835 treatise Sur les équations du mouvement relatif des systèmes de corps, extended Newtonian mechanics to rotating reference frames by incorporating an additional acceleration term—now known as the Coriolis acceleration—derived in units consistent with metres per second squared, applied to phenomena like machinery and fluid motion. This work implicitly relied on metric-derived units for precise kinematic descriptions, predating the International System of Units (SI) by over a century.[43] The unit's status was further solidified in 1889 when the International Bureau of Weights and Measures (BIPM), established by the Metre Convention of 1875, sanctioned the first international prototype metre—a platinum-iridium bar—during the 1st General Conference on Weights and Measures (CGPM). This artifact defined the metre internationally, enabling consistent derivation of m/s² as a coherent unit for acceleration, independent of named force units like the newton (formalized later in 1948). While acceleration measurements in this era often used centimetre-gram-second (CGS) variants, the m/s² remained the foundational metric expression in theoretical mechanics.[44]

Evolution and Standardization

The formalization of the metre per second squared (m/s²) as the coherent SI unit of acceleration began with the establishment of the International System of Units (SI) at the 11th General Conference on Weights and Measures (CGPM) in 1960. Resolution 12 of this conference adopted the name Système International d'Unités and defined a coherent set of units based on the metre, kilogram, and second, with acceleration expressed as m/s² to ensure dimensional consistency in mechanical equations.[45] This marked a shift toward a unified framework for scientific measurements, building on earlier metric foundations but emphasizing practicality for international use. Subsequent refinements stabilized the unit's metrological basis. In 1967, the 13th CGPM redefined the second via Resolution 1 as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom, replacing ephemeris time standards and providing an atomic reference that enhanced precision for time-dependent units like m/s².[46] The 17th CGPM in 1983 further anchored the metre through Resolution 1, defining it as the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second, which fixed the speed of light at exactly 299,792,458 m/s and thereby stabilized m/s² against material artifacts.[47] Additionally, the 13th CGPM's Resolution 6 explicitly outlined SI derived units, confirming m/s²'s role without special naming, while the 15th CGPM in 1975 extended this by adopting special names for other derived units (e.g., gray for absorbed dose), reinforcing the system's coherence.[48] The 26th CGPM in 2018, effective from 20 May 2019, completed this evolution by redefining all SI base units in terms of fixed numerical values of fundamental constants, eliminating reliance on physical prototypes. Resolution 1 fixed the caesium hyperfine transition frequency and the speed of light, ensuring m/s²'s invariance and universality without further adjustments.[49] The BIPM's 8th SI Brochure (2006), updated in the 9th edition (2019), formalized the notation as m s^{-2} and emphasized its coherence within the revised system, where acceleration derives directly from base units without dimensional factors.[3] Global adoption accelerated in the 1970s, as international bodies and scientific publishers mandated SI units in literature, supplanting the centimetre-gram-second (CGS) system's cm/s², particularly in physics where CGS had dominated electromagnetic and mechanical contexts. By the mid-1970s, major journals and organizations, including those affiliated with the American Physical Society, required m/s² for consistency, facilitating cross-disciplinary collaboration and standardizing experimental reporting worldwide.

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