Metre per second squared
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 = \frac{\Delta v}{\Delta t}, where a is the acceleration in m/s², \Delta v is the change in velocity, and \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 + \frac{1}{2}at^2, where s is the displacement, u is the initial velocity, t is the time elapsed, and a 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 F acting on an object equals its mass m times its acceleration a: 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 isg(\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](/page/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 = \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 (a_x = 0), while the vertical component is directed downward (a_y = -g), with g 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 a_c = \frac{v^2}{r}, where v is the tangential speed in m/s and r 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 $10^{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 \, \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 \, \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 \, \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 \, \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 \, \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 \, \mathrm{kn/s} = 0.514444444 \, \mathrm{m/s^2},
so
$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]
| Unit | Symbol | Conversion from m/s² | Notes |
|---|---|---|---|
| Foot per second squared | ft/s² | ≈ 3.28084 ft/s² | Derived from 1 ft = 0.3048 m exactly |
| Galileo | Gal | = 100 Gal | CGS unit; 1 Gal = 1 cm/s² |
| Standard gravity | g | ≈ 0.10197 g | g = 9.80665 m/s² exactly by definition |
| Centimetre per second squared | cm/s² | = 100 cm/s² | Direct CGS base unit |
| Knot per second | kn/s | ≈ 1.94384 kn/s | 1 kn = 1852/3600 m/s exactly |