G-Force Calculator: Acceleration in g

G-Force Calculator

A g-force is an acceleration written as a multiple of gravity at the Earth's surface. A turn described as "3 g" has an acceleration three times standard gravity. This calculator converts an acceleration into g and back, and works one out from a change in speed or from a turn.

What the calculator does

The page has three modes. All three end at the same place: one acceleration, shown at once in metres per second squared, feet per second squared and g.

A speed may be entered in metres per second, kilometres per hour, miles per hour, feet per second or knots. A radius may be entered in metres, kilometres, centimetres, feet, yards or miles. The site's shared unit registry converts every entry before the arithmetic runs, so the foot used here is the same foot used on every other page.

Results are shown to eight significant figures. Only the display is rounded. The arithmetic runs on the values as typed. A result below 0.0001, or 1,000,000,000 and above, is shown in scientific notation, and a result of zero is shown as a plain 0.

One g, and the g-force formula

One g is standard gravity, written $g_n$. Turning an acceleration into g is a single division:

$a_g = \frac{a}{g_n}, \qquad g_n = 9.80665\ \mathrm{m/s^2}$

where $a$ is the acceleration in metres per second squared and $a_g$ is that same acceleration counted in g. Multiplying by $g_n$ goes back the other way.

Standard gravity is a convention rather than a measurement of any particular place. Real gravity is a little weaker at the equator and on a mountain, a little stronger at the poles. The 3rd General Conference on Weights and Measures fixed the single agreed value in 1901, and NIST SP 811 Appendix B tabulates it, so the number is exact and will not be revised by a better experiment.

Feet per second squared uses the foot of the 1959 International Yard and Pound Agreement:

$a_{\mathrm{ft/s^2}} = \frac{a}{0.3048}$

where 0.3048 is the number of metres in one foot, exactly.

How to calculate g-force from a change in speed

$a = \frac{\Delta v}{\Delta t}$

where $\Delta v$ is the final speed minus the starting speed, in metres per second, and $\Delta t$ is how long the change took, in seconds. The answer is an average across the whole interval rather than a peak.

A change in speed may be negative. Slowing down is negative acceleration, so the page keeps the sign and says beside the numbers whether the body was speeding up or slowing down.

How to calculate g-force in a turn

$a = \frac{v^2}{r}$

where $v$ is the speed around the circle in metres per second and $r$ is the radius of the circle in metres. This is centripetal acceleration, the inward pull that holds a body on a circular path at a steady speed. It grows with the square of the speed, so doubling the speed around the same circle quadruples the g-force, and it falls as the circle widens.

Squaring removes the sign of the speed. That is correct rather than convenient, because the pull is inward whichever way around the circle the body runs.

Which constants are exact

Every number these conversions rest on is a definition agreed by a standards body, not a measured quantity.

One figure that looks like a constant is not one. Gravity in imperial units is often quoted as 32.17405 feet/s², and that is a rounding. The true ratio of 9.80665 to 0.3048 has a decimal expansion that never ends, running 32.174048556430446 and onward, so any decimal written for it is cut short somewhere. Entering 32.17405 feet/s² on this page returns 9.8066504 m/s² rather than 9.80665 m/s². Both read 1 g at eight significant figures, because they first differ in the ninth.

Worked example

A car reaching 100 km/h in 4 seconds. The change in speed is 27.777778 m/s. Divided by 4 seconds that is 6.9444444 m/s², which the page shows as 22.78361 feet/s² and 0.70813626 g.

A car reaching 60 mph in 6 seconds. Because a mile is exactly 1,609.344 m, 60 mph is exactly 26.8224 m/s. Divided by 6 seconds that is exactly 4.4704 m/s², which is 14.666667 feet/s² and 0.45585394 g.

A car through a 50 m corner at 200 km/h. The speed is 55.555556 m/s. Squaring it and dividing by the 50 m radius gives 61.728395 m/s², which is 202.52098 feet/s² and 6.2945445 g.

Free fall. A body falling with nothing but gravity acting on it accelerates at 9.80665 m/s², which is 32.174049 feet/s² and exactly 1 g.

How the arithmetic is arranged

Eight significant figures is wide enough to show rounding dust, so the calculation is arranged to cancel exactly at the points where a reader expects a round answer. Entering 9.80665 m/s² shows 1 g, entering 1 g shows 9.80665 m/s², and entering 1 foot/s² shows 1 foot/s² again, with nothing trailing.

The speeding-up or slowing-down label is decided on a rounded value rather than a raw one. Computers store these numbers in binary, and a change of speed that ought to be zero often is not: 0.1 plus 0.2 minus 0.3 comes out as 0.000000000000000055511151. A label read from the raw sign would say "speeding up" while the card beside it printed that residue as an acceleration. Anything that rounds to zero at twelve decimal places is therefore reported as no change at all, in the label and in the numbers alike, so the two cannot disagree. Twelve places is far below anything a reader enters here. The Pioneer anomaly, about the smallest acceleration anyone quotes a figure for, is 0.00000000087 m/s², more than a thousand times larger than the cut-off.

What the calculator refuses

An input box passes through whatever is typed or pasted into it, and a shared link can carry any value at all, so the checks that matter live in the calculation code rather than in the form.

Negative accelerations are not refused. Acceleration is signed, and slowing down is the ordinary case for the negative half.

What the figure is not

It is the acceleration itself, not what a body feels. Someone standing still already reads 1 g on an accelerometer, because the ground is pushing up on them, and this page does not add that baseline to the acceleration entered.

The change-in-speed mode reports an average over the time given. A short event such as a crash peaks well above its average, so an instrument sampling the same interval reads higher. The turn mode assumes a steady speed, so it counts only the inward part of the acceleration. Speeding up or slowing down while turning adds a part along the path that this figure leaves out. The arithmetic is Newtonian, with no relativistic correction.

Frequently asked questions

What is a g-force? It is an acceleration written as a multiple of standard gravity. One g is 9.80665 m/s², so 3 g is an acceleration three times that size. Despite the name it is an acceleration rather than a force, and it has no unit of its own, because it is one acceleration divided by another.

How do you convert m/s² to g? Divide by 9.80665. An acceleration of 19.6133 m/s² is 2 g. To go the other way, multiply the figure in g by 9.80665.

How many g is 0 to 60 mph in 6 seconds? About 0.46 g. 60 mph is 26.8224 m/s, dividing by 6 seconds gives 4.4704 m/s², and dividing that by 9.80665 gives 0.45585394 g. That is the average over the six seconds, and the peak partway through is higher.

How is the g-force of a turn worked out? Square the speed in metres per second, divide by the radius in metres, then divide by 9.80665. At 55.555556 m/s around a 50 m circle the acceleration is 61.728395 m/s², or 6.2945445 g.

Is 32.17405 feet/s² one g? Not exactly. It is a rounding of a ratio whose digits never end. Entering it here returns 9.8066504 m/s² rather than 9.80665 m/s². Both still read 1 g at eight significant figures.

Why does a negative change in speed give a negative answer? Because acceleration carries a direction. A drop in speed is negative acceleration, and reporting it as a positive number would lose the fact that the body was slowing down.

How many g can a person take? That depends on the direction of the acceleration and on how long it lasts, and it varies from person to person. A few g held for a while is enough to cause trouble for an untrained person, while a very short spike of far more can be survivable. This page reports the acceleration and leaves the verdict alone. Treat any figure here as physics, not as medical advice.