Jerk Physics Calculator

What Is the Jerk Physics Calculator?

How to use the jerk physics calculator

1. Choose the unknown. The Solve for menu opens on Displacement. The other three choices are Jerk, Initial acceleration and Initial velocity, and whichever you pick vanishes from the boxes below.
2. Type the initial velocity. Initial velocity is how fast the body was already going when the clock started, in m/s or km/h. Leave it at zero for anything starting from rest; its term is the only one that grows in simple proportion to the time.
3. Type the initial acceleration. Initial acceleration is the acceleration at the very start, in m/s2 or multiples of g, and it is the single figure a constant-acceleration equation would use for the whole interval. Setting it to zero means the body is not yet pulling at all.
4. Type the jerk. Jerk is how fast that acceleration is changing, in m/s3, and it is the box that makes this page different from every other kinematics tool here. Positive values make the acceleration climb through the interval and negative ones bring it back down.
5. Set the elapsed time. Elapsed time is the length of the interval the jerk is held over, in s or min, and it is an input in every mode rather than an answer. It is also the most powerful box on the page, because the term the constant-acceleration formula drops grows as the cube of it.
6. Read the answer and the chips. The headline is the quantity you asked for, to four significant figures. The chips give the final velocity, the final acceleration, the mean acceleration, what SUVAT would have said, the term SUVAT drops and the trapezium estimate.
7. Compare the answer with the SUVAT chip. On the opening case the chip reads 24.000 m against an answer of 40 m, so a constant-acceleration equation misses 16.000 m of the forty. Load Lift, acceptable jerk and the chip reads 0.000 m against 2.667 m, which is the comparison this page exists for.
8. Open Show working. The steps restate the relation, list your figures in SI base units whatever the menus say, and then print the three terms separately. Seeing 1 min come back as 60 s in that second line is the quickest way to see what the conversion did.

This page starts where a constant-acceleration tool stops. For the five equations themselves, the symbol table and the rule that the acceleration must hold still, the guide to the SUVAT equations covers all of it and this page assumes it. The SUVAT calculator is the right tool whenever that rule holds, and it has no box for a jerk because it does not need one.

That guide also gives the correct practical fix when an acceleration changes: split the motion into stages that each hold their own. Nothing here contradicts it, and for a stepped or irregular acceleration it is still the right answer. What this page adds is the one case where no splitting is needed, because the acceleration ramps at a steady rate and the whole interval closes in a single term.

Two mistakes account for most wrong answers here, and both are about which acceleration was typed. The first is entering the acceleration measured at the end of the interval in the Initial acceleration box, which overshoots rather than undershoots; the final acceleration chip tells you what that figure should have been. The second is a time typed in minutes while the menu still reads seconds, which the second line of Show working will always catch.

A third is subtler, because the box takes the figure without complaint. If the acceleration you have is an average between two velocities rather than the value it started at, the guide to acceleration in physics sets out the difference, and the acceleration calculator is the tool that works that average out.

Worked example: change one thing at a time

The table starts at the defaults and moves one thing at a time: which quantity is the unknown, then the case, then the jerk, then the time, then the unit the figures are typed in. Every Result cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool. The input column is simply what you type, in the order the boxes appear.

Rows 1 to 4 are one state read four different ways. Each answer returns the figure the row above it started from, so the displacement gives back the jerk, then the initial acceleration, then the initial velocity. Four unknowns, one relation, and no arrangement more fundamental than another.

That round trip is the genuine check on this page, and it is worth saying why the last column is not. In row 1 the answer is the SUVAT figure plus the dropped term, so the two agreeing is the relation restated. Rows 2 to 4 do different arithmetic in the other direction and still land on 1.5 m/s3, 3 m/s2 and 0 m/s, which is a claim that can fail.

Rows 5 to 7 are the demonstration this page exists for, and they are not a contrived case. All three start from rest with no acceleration either, so both terms a constant-acceleration equation keeps are zero and it predicts the lift never moves. The lift has in fact risen 2.667 m, 0.9333 m and 8 m; state that in metres, because dividing by the 0.000 m in the last column is a division by zero rather than a percentage.

Those three rows share an initial velocity, an initial acceleration and a time and differ only in the jerk, so their displacements stand in exactly the ratio of their jerks. That is arithmetic rather than a finding, and it is worth noticing only because it makes the third term's role obvious. Row 8 runs the last of them backwards and recovers 6 m/s3 from the 8 m it produced.

Rows 9 and 10 turn the sign around. A negative jerk brings the acceleration down through the interval, so the constant-acceleration figure overshoots instead: 36.000 m against a true 30.67 m for the cruise-like ramp. Row 10 recovers 20 m/s from a braking-like case in which both the acceleration and the jerk are negative.

Rows 11 and 12 are the two ends of the argument. Set the jerk to zero and the answer falls to the SUVAT figure exactly, which is a definition rather than a discovery; double the time and the dropped term climbs from 16.000 m to 128.000 m, which is eight times as much for twice the clock. Rows 13 to 16 are the edges — 72 km/h and 1 g come back as the same answers their SI equivalents give, one minute of the rated acceptable jerk reaches 72000 m, and a time of zero is refused rather than answered.

Formula and symbol reference

The calculator uses one relation in four arrangements. The displacement is s = u t + 1/2 a0 t2 + 1/6 j t3, so the jerk is j = 6 (s - u t - 1/2 a0 t2) / t3, the initial acceleration is a0 = 2 (s - u t - 1/6 j t3) / t2 and the initial velocity is u = (s - 1/2 a0 t2 - 1/6 j t3) / t.

Each of those four appears linearly, which is why each has exactly one answer and why all four are on the Solve for menu. The elapsed time does not: it appears as a cubic, and a cubic has up to three real roots with no honest way for a calculator to choose between them. That is the reason the time is an input in every mode, and it is a limitation worth stating rather than hiding.

The physics: the one term a constant acceleration has nowhere to put

Write the position as a series about the start of the interval and the first term is the initial velocity times the time, the second is half the initial acceleration times the time squared, and the third is a sixth of the jerk times the time cubed. A constant-acceleration equation keeps the first two and has no third. That is not a flaw in it; it is what "constant acceleration" means.

When the jerk really is constant, that third term is not an approximation either. The series stops there, because the fourth derivative of the position is zero, so s = u t + 1/2 a0 t2 + 1/6 j t3 is exact for this motion rather than a better estimate of it. The chips beside the answer are then exact too, which is why the final acceleration chip reads a flat 0.000 m/s2 on the cruise-like ramp.

Three terms, three different growth rates — that is the whole of the practical argument. The velocity term grows as the time, the acceleration term as its square, and the dropped term as its cube. Over a short interval the third is negligible; double the interval and it multiplies by exactly eight, which the table above shows as 16.000 m becoming 128.000 m.

There is an exact moment when the dropped term catches the one a constant-acceleration equation keeps: the jerk term equals the acceleration term at t = 3 a0 / j. On the opening case that is 6 s, and the calculator is set to 4 s, which is why the 16.000 m chip is already two thirds of the 24.000 m one. The companion crossover with the velocity term is at t = sqrt(6 u / j).

Neither of those is a universal constant, and it matters that no page treats them as one. Each depends on the other two figures you typed, so "the time when SUVAT stops working" is not a number that exists. The honest statement is that for these inputs the two terms are equal at that moment, and you can watch the 16.000 m chip pass the 24.000 m one by typing a larger time.

The trapezium estimate chip is there because it is the repair most people reach for and it is not safe here. Half the sum of the starting and finishing velocities, times the time, is exactly what a constant-acceleration equation gives if you feed it the mean acceleration chip — one formula wearing two names — and both are long by a twelfth of the jerk times the time cubed. The reason is visible on a velocity-against-time graph: when the acceleration ramps, that line is a parabola, and a trapezium cannot measure the area under a parabola.

That area is the displacement, which is the method the guide to motion graphs already teaches and which remains correct here. It is only the straight-line shortcut for evaluating it that breaks, and the motion graphs simulator is the place to see the straight version of that line before this page bends it.

Where the jerk physics calculator breaks down

The arithmetic is short and hard to get wrong. What fails is the relation being pressed onto a motion it does not describe, or a published figure being carried somewhere it does not belong.

Constant jerk is itself an idealisation

The Jerk box holds one number for the whole of Elapsed time, and in real motion that number moves as well. Past the cubic term the page computes there is a quartic one, the fourth derivative of position divided by twenty-four and multiplied by the time to the fourth power, and no box here will take it. So every answer is one term further on than a constant-acceleration result rather than a final one, exact only where the jerk you typed really does hold steady, and the longer the interval you ask for the further out it will sit.

One set of boxes covers one ramp

The five fields describe a single interval over which one jerk applies, so a journey that changes character part way through is not one sum on this page. Run it as two or three passes: read the final velocity and final acceleration chips off the first stage, type them in as the second stage's starting figures, and add the displacements yourself. All the page spares you is the splitting a steadily ramping acceleration would otherwise need.

The model is one-dimensional

There is one Displacement box and no direction to go with it, so the answer is a signed position change along one line and not a distance covered. Something that travels out and returns is entitled to a small figure here. Motion that also turns has a jerk with components these boxes cannot hold, and nothing on the page will warn you that you have typed one.

The published rail figures are lateral, not longitudinal

A high-speed rail design range of 0.2 to 0.6 m/s3 describes the jerk caused by track curvature changing under a train, which acts across the direction of travel. This page's model runs along the direction of travel, so those numbers belong in the conversation as the magnitude engineers design to and not in the jerk box.

The lift figures are rated jerks, not ride profiles

Press Lift, intolerable jerk and that rated figure, held across the whole two seconds the case asks for, arrives at 12 m/s2 and 12 m/s — around forty-three kilometres an hour straight up, which is not what any lift in service does. A real machine spends a fraction of a second ramping its acceleration and then keeps it there, so for most of a ride the honest entry in the Jerk box would be zero. Take what the three lift buttons return as the arithmetic of a rated jerk held far longer than anybody holds it.

No standard sets a jerk limit for lifts

ISO 8100-34 is a measuring standard rather than a limiting one: what it fixes is the method by which a lift's ride quality is recorded — jerk among acceleration, vibration and noise — and it does not specify acceptable levels, or unacceptable ones either. So the 2, 6 and 0.7 m/s3 the preset buttons drop into the Jerk box are reported passenger ratings and one hospital recommendation, with nothing official beneath them. This page can tell you what any of the three does over your interval; it cannot tell you whether a standard permits it.

Two of the chips are the same subtraction, and two are the same formula

The term SUVAT drops and the gap between the answer and the SUVAT chip are one calculation printed twice, so their agreeing confirms nothing. The trapezium estimate and a constant-acceleration equation fed the mean acceleration chip are likewise one formula under two names. Watching the answer drop onto the SUVAT figure once the jerk is set to zero is a definition, not a test.

The elapsed time is never the answer

Ask how long something took and the relation becomes a cubic with up to three real roots, more than one of which can be physically sensible. Rather than choose one silently, this page leaves the time as an input in all four modes. If you need it, sweep the time box and watch the displacement chip cross the figure you are aiming at.

Rounded figures in, rounded figures out

The headline carries four significant figures and the chips three decimals, while the arithmetic behind both is unrounded, so the printed terms need not agree with the printed total in their last digit. Show working is the place to look instead: it repeats the answer to as many as seven significant figures and pads none of them, so the hospital case reads 0.9333333 there against a headline of 0.9333, while the opening case prints a plain 40 in both places. Typing a rounded displacement back in as an input has the matching effect, and the jerk that comes out sits a little off the one the case began with.

The calculator measures nothing — you supply every figure

Five boxes, converted to SI base units and put through one relation: that is the whole of what happens here, so an answer can only be as sound as the numbers typed above it. A headline describes your inputs and not any lift, vehicle or journey, and the presets are there to show the relation working rather than to report a real machine. Verify anything you mean to rely on against your own data before you quote it.

Where jerk is actually used

Judging lift ride quality

Passenger comfort in a lift is governed by how quickly the acceleration changes rather than by how large it gets, which is why jerk has its own reported figures at all: a vertical jerk of 2 m/s3 is reported as acceptable to most passengers and 6 m/s3 as intolerable, with 0.7 m/s3 recommended for hospitals. Load the three lift cases to see what each of those does to two seconds of travel: 2.667 m, 8 m and 0.9333 m respectively.

Shaping transition curves on rail and road

A curve that begins abruptly forces a step change in sideways acceleration, so track and carriageway alignments use a transition whose curvature builds gradually and whose design target is a jerk rather than an acceleration. Published design ranges for high-speed rail run from 0.2 to 0.6 m/s3, measured across the direction of travel rather than along it, so they set the magnitude engineers work to rather than a number to type into this page.

Turning a measured acceleration ramp into a jerk

If you know how far something moved and what it started from, the jerk that produced it is the answer this page gives with the Solve for menu on Jerk. That is exactly what the Lift, hospital limit, back to the jerk case sets up: a rise of 0.9333333 m in two seconds from rest returns 0.7 m/s3. The same mode turns a recorded stopping distance into the rate at which the braking was eased off.

Sizing the error before trusting a constant-acceleration answer

Arguing about whether an acceleration is constant enough gets nowhere; the calculator turns it into a subtraction. Put the interval you actually care about in Elapsed time and read the term SUVAT drops beside the headline, because that chip is the whole of what a constant-acceleration equation would miss, in metres. On the opening case it prints 16.000 m against an answer of 40 m, so the shortcut is out by well over a third; shorten the interval and the chip falls away far faster than the answer does.

Recovering the state a stage began in

A journey broken into stages leaves you knowing where each stage ended but not always what it began with. With the menu on Initial velocity or Initial acceleration, the same relation runs backwards and hands that starting figure back, which is what the two cases named "back to the start" and "back to the velocity" demonstrate. Both are ordinary uses of the tool rather than special cases.

Reading the final acceleration chip before quoting an acceleration

When the acceleration is ramping there is no single value of it, so a figure quoted without saying when it was measured is ambiguous. The chips give you all three at once: the value you typed at the start, the final acceleration at the end of the interval, and the mean acceleration between them. On the opening case those are 3, 9.000 and 6.000 m/s2, and quoting the wrong one of the three is the commonest way to get a wrong answer out of a right equation.

Where to go next

For the method in full, with worked problems and the diagrams that go with them, read Jerk in Physics: Displacement When Acceleration Changes. For the constant-acceleration world this page sits just outside — the five equations, the symbol table and the rule that decides which one to use — What Are the SUVAT Equations? is the place to go, with the SUVAT calculator beside it.

Two more are worth a bookmark. The acceleration calculator handles a mean acceleration on its own, and Motion Graphs in Physics covers the area rule this page leans on. The whole physics lab library is open too, if you would rather watch a curve than type one.