Tangential Acceleration Simulator

What Is the Tangential Acceleration Simulator?

How to use the tangential acceleration simulator

1. Start from the state it opens in. The lab boots on Speed v 15.0 m/s, Bend radius r 45 m, Tangential acceleration a_t 0.0 m/s2 and Grip coefficient μ 0.80, with Centripetal, inward and Total acceleration both reading 5.000 m/s2. Reset returns all four sliders to exactly that and stops the motion if it is running.
2. Set the speed. Speed v runs from 0 to 40 m/s in steps of 0.5 and is squared inside the inward part, so it is the slider the inward arrow answers to most sharply. Take it to 0.0 m/s and that arrow has no length left to draw, which is a state the panel still reports in full.
3. Set the bend radius. Bend radius r runs from 5 to 200 m in steps of 1 and divides the inward part, so a tighter bend at the same speed raises it. The drawn circle keeps its size on screen because the path is always scaled to fit; the radius you chose is printed under the canvas instead.
4. Set the tangential acceleration. Tangential acceleration a_t runs from −8 to 8 m/s2 in steps of 0.1, negative for braking and positive for powering out, and it is the slider that makes this lab different from the circular motion lab. Moving it off zero is what brings the second solid arrow, the dashed resultant and the angle marker onto the canvas together.
5. Set the grip coefficient. Grip coefficient μ runs from 0.2 to 1.5 in steps of 0.05 and changes no acceleration at all: the total stays 6.403 m/s2 on the braking case whether it sits at 0.80 or 0.35. What it moves is the Grip budget μg cell, from 7.848 m/s2 to 3.433 m/s2, and with it the whole scene’s scale.
6. Read the four cards down the right. Centripetal, inward carries its own formula line, a_c = v^2/r; Total acceleration carries sqrt(a_c^2 + a_t^2); Angle off the radius gives the lean in degrees; and What the tyres are doing prints one of exactly four phrases, from Not moving yet to Grip exceeded.
7. Read the nine cells under the canvas. They give the Tangential term a_t, the Grip budget μg, the Budget in use, the Fastest at this a_t and Fastest at a steady speed pair with the Ratio of the two, the Speed where the lean hits 45°, and the Angular velocity ω and Angular acceleration α.
8. Read the strip headed The drawing. It describes the canvas in words and switches between three forms as the scene changes, so it can never claim a lean that is not on the screen or stay silent about one that is.
9. Press Play to let it run. The object travels round the bend and Speed v evolves at the tangential acceleration you set, clamped to the slider range, while the button relabels itself Pause. Nothing else moves, which is what keeps every reading a function of the four slider positions alone.

The dashed arrow is the reading to watch. It carries both components at once, and the Total acceleration and Angle off the radius cards are its size and its direction put into figures. Work the third slider first and the grip slider after it: the first moves both of those cards, the second moves neither.

If the speed in your problem really is constant, this is more lab than you need. The guide to circular motion physics covers that case in full — the definition, the symbol table and the three equivalent forms — and everything it says stays true here, because the inward arrow is exactly the acceleration it describes. Come back the moment the speed starts changing.

Worked example: change one thing at a time

Every row below is one of the six preset buttons, and every cell is a string the running lab printed there. Rows 1 and 2 hold the speed, the bend and the grip setting still and move only the tangential slider; rows 4 and 5 take the speed away; row 6 repeats row 2 with the grip setting cut. Where a cell and the lab ever part company, believe the lab.

Rows 1 and 2 are the whole lesson. Same bend, same speed, same grip setting, and the inward part does not move at all: 5.000 m/s2 in both. What changes is the total, which climbs to 6.403 m/s2, and the direction, which swings 38.7 deg away from the centre. Nothing has gone wrong with the inward figure — it has simply stopped being the whole answer.

Row 4 is the case a constant-speed formula returns nothing for. At a standing start the inward part really is 0.000 m/s2, so a = v2/r is telling the exact truth about the component it describes, and the whole 2.500 m/s2 lies along the path. Read the 90.0 deg from the card rather than from the drawing: with no inward arrow to measure from, that scene marks no angle at all.

Two cells in the table are words rather than numbers, on purpose. Row 5 prints the exact string undefined for the angle, because with nothing moving and nothing changing there is no direction to report and zero degrees would be a plausible-looking lie. Row 6 prints no speed works in the limit cell, because the braking alone has spent the whole budget before any turning.

Neither of those is a number and nothing on this page divides by either. If you want the quantities that are numbers to more decimal places, or the same relation run backwards from a total you already have, the tangential acceleration calculator takes typed input and solves it four ways.

Rows 2 and 6 are one manoeuvre at two grip settings. The total is 6.403 m/s2 in both, because the total does not depend on the grip coefficient at all; only the budget does, falling from 7.848 to 3.433 m/s2 and taking the budget in use from 81.6 % to 186.5 %. The manoeuvre is identical and the verdict is not.

Those two word-strings are different thresholds and must not be read as one. Grip exceeded compares the total against the budget, which is the same comparison as the budget in use passing 100 %, so the two appearing together is a convenience rather than a second opinion. no speed works says something stronger: the speed change alone has used the whole budget, so no cornering speed fits at any radius.

Formula and symbol reference

The lab works from one relation, a total = sqrt(a_c2 + a_t2) with a_c = v2/r, and reports the lean as φ = atan(a_t / a_c). The grip cells add one constant to that: the budget μg with g taken as 9.81 m/s2 throughout.

Because the two components meet at right angles, the first slice of speed change is nearly free. Hold the opening bend and walk the tangential slider up from zero: Total acceleration reads 5.000 m/s2 at 0.0 m/s2, 5.004 at 0.2, 5.009 at 0.3 and only 6.403 once the braking reaches 4.0. The cost climbs steeply only when the two components become comparable in size.

Two rows there are easy to misread. The Ratio of the two cell is worked out from the unrounded speeds, so it prints 0.9276 on the braking case while dividing the two printed strings, 17.431 by 18.793, gives 0.927526: they part company in the fifth decimal, and the cell is the one to quote. The Angular velocity ω and Angular acceleration α cells are the speed and the tangential term divided by the radius, so they restate the same state on the rotation side rather than confirming it — the guide to the angular velocity formula works that pair out properly.

The physics: why the acceleration stops pointing at the centre

The claim this lab makes is a claim about a drawing. The solid inward arrow is the acceleration every other circular-motion tool here draws, the second solid arrow is the one they leave out, and the dashed arrow between them is what the object has. Once both solid arrows are on screen, the three of them form a right-angled triangle in pixels, and the dashed one is genuinely the longest side of it.

That triangle is why a little speed change is cheap. Perpendicular quantities add through their squares rather than end to end, so 0.3 m/s2 of braking on top of 5.000 m/s2 of turning moves the total by nine thousandths, while 4.000 m/s2 moves it by 1.403. The angle is the livelier reading of the two: that same 0.3 m/s2 already leans the total 3.4 deg off the radius, and the drawing shows a direction far better than it shows a length.

The dashed circle is the second half of the argument. It is the grip budget drawn to the same ruler as the arrows, so a resultant that reaches past it is a manoeuvre asking for more grip than the setting supplies. Load Powering out too hard and the tip crosses the circle while the Budget in use cell reads 119.5 %: the picture and the number are saying the same thing, because the circle is drawn from that same budget.

The ruler is rebuilt for every scene, and that is the one thing you must not read across. The scale is whatever makes the longer of the resultant and the grip radius exactly 86 px, so Steady bend, Braking into the bend, Pulling away from rest and Stopped on the bend all draw at 10.9582 px per unit of acceleration, while Powering out too hard draws at 9.1670 and Low-grip setting at 13.4309. Inside any one scene every length is comparable; across that boundary, none of them is.

The first two shots on this page are both inside that group of four, which is why they can be read against each other. The inward arrow is 54.79 px in the steady scene and 54.79 px in the braking one, exactly as the unchanged 5.000 m/s2 card says it should be, and the 43.83 px tangential arrow is what has been added. The same reading across the boundary would mislead: at Powering out too hard an 8.889 m/s2 inward arrow draws at 81.48 px against that 54.79, a drawn ratio of 1.487 where the accelerations stand in the ratio 1.778, which is why this page compares no length between those scenes.

There is an exact speed at which the two components are equal, and the Speed where the lean hits 45° cell prints it: 13.416 m/s on the braking case, 7.071 m/s on the 20 m bend at 2.500 m/s2. The speed slider steps in halves, so the nearest stop to the first is 13.5 m/s, where the angle card reads 44.6 deg. It is not a universal constant: it moves with both the radius and the tangential term, and it does not depend on the speed the object started at.

The pair of speed cells is where the grip argument pays off. On the braking case Fastest at this a_t reads 17.431 m/s against 18.793 m/s at a steady speed — the ceiling has fallen by about seven per cent, and nothing has slowed the object down. Drag the grip slider to 0.65 and the tyre card turns over to Grip exceeded at 100.4 %, while the total stays 6.403 m/s2 throughout.

None of this is a statement about force. The lab reports accelerations only, and turning one of them into the force behind it needs the mass, which is a step these four sliders never take — the guide to centripetal force is where that difference is set out properly.

Where the model breaks down

The lab solves its own model exactly, so nothing on the screen ever fails. Everything below is a limit of that model, of the figures fed into it, or of what the drawing can carry, and each item says what this lab does about it.

The grip coefficient is a figure you choose

Nothing in this lab measures a surface. The slider is a number you choose, and the presets are named after the manoeuvre for exactly that reason: Low-grip setting is a setting rather than a wet road. Quoting 0.80 or 0.35 as the coefficient of any real road, tyre or vehicle would be inventing a measurement this page has not made.

The grip circle is itself an idealisation

Treating the limit as one budget shared between turning and changing speed draws a perfect circle, and a real tyre limit is not a perfect circle. It changes with the load the tyre carries, is rarely the same size in both directions, and depends on the surface and the tyre together. The circle here is a teaching shape that makes the trade-off visible, not a measured envelope.

Arrow lengths cannot be compared between two scenes

The scale is rebuilt for every setting so that the longer of the resultant and the grip radius fills the space available, which keeps the drawing legible and makes a length meaningful only inside one scene. Four of the six presets happen to share 10.9582 px per unit; the other two do not. Read lengths against each other within a scene, and read magnitudes off the cards.

An arrow vanishes at a pixel width, not at a physical threshold

Anything shorter than three pixels on the canvas as drawn is not drawn at all, its label goes with it, and the strip switches form in the same pass. On the opening bend the tangential arrow survives at 0.3 m/s2 and disappears at 0.2, where the angle card goes on reading 2.3 deg quite correctly. That threshold moves with the canvas width, so it is a fact about the picture.

Every reading is one instant, not a journey

The four sliders describe one moment rather than a journey. The tangential acceleration is the rate at which the speed is changing now, and nothing integrates it forward except Play, which moves the object and the speed slider and leaves every other quantity a function of where the sliders are. Treat a reading as one frame out of many.

One radius, held for the whole reading

The bend drawn here is a circular arc: the second slider fixes its radius and the scene keeps it, so a corner that closes up as it goes, or an entry whose curvature is still building, is outside what these four controls can describe. Step the radius down a few metres at a time and read the cards at each stop if you want to see where such a path is heading; what the lab cannot give you is one reading covering the whole sweep.

The sign of the tangential slider never reaches the cards

The total and the angle depend on the size of the tangential term and never on its sign, so the two are indistinguishable to every reading on this page. Set the slider to 4.0 m/s2 instead of −4.0 and the cards do not move. Where braking mid-bend really is the classic mistake, the reason lies in when it is done rather than in the physics charging more for it.

Two of the readings are the same comparison seen twice

The Budget in use cell passing 100 % and the tyre card reading Grip exceeded are one comparison printed in two forms, so watching them agree confirms nothing. The same holds for the total being the hypotenuse of the two components, which is the definition of the total, and for the lean reaching 45 deg, which is the same statement as the two components being equal.

Rounded cells, unrounded arithmetic

Every card carries a fixed number of decimals and the calculation behind it carries none of that rounding, so two printed figures need not reproduce a third in its last digit. The ratio cell is the honest example: it prints 0.9276 while the two speeds it came from print as 17.431 and 18.793, whose quotient is 0.927526. Take ratios from the cell, not from the strings.

A steady speed is still the ordinary case

Set the tangential slider to 0.0 m/s2 and this lab collapses into the one every other circular-motion tool on the site draws, with the total equal to the inward part and the acceleration aimed at the centre. Everything the circular motion guide says about that case remains correct; this page adds a term rather than replacing one.

The bend is drawn flat, and a real one need not be

Everything on this canvas lies in one horizontal plane, which is what puts the inward arrow and the arrow along the path at right angles to begin with. Bank or camber the surface and some of the turning is done by the tilt, gravity gains a component the scene never draws, and the dashed circle stops being the comparison that matters. The banked curve lab draws a tilted bend properly.

Nothing here has been measured

All four sliders are figures you chose, and no reading on this page describes any vehicle, surface or journey. Take the speed to 40.0 m/s on the tightest bend and the lab reports what that implies without comment on whether anything does it. Verify any figure you intend to rely on against your own data first.

Where tangential acceleration is actually used

Seeing the friction circle instead of arguing about it

The idea that turning and changing speed share one budget is easy to state and hard to picture. Here it is a dashed circle with a dashed arrow in it: park the tip near the edge, then move the tangential slider and watch the tip swing out through the circle while the inward arrow never moves. That is the trade-off drawn rather than asserted.

Judging how much a speed change really costs

The useful question is never whether the speed is changing but whether it changes enough to matter. Put the real figure into the tangential slider and read Fastest at this a_t against Fastest at a steady speed: 17.431 m/s against 18.793 m/s at 4.000 m/s2 of braking. That is an answer about the ceiling in metres per second, not an argument in principle.

Laying out the entry to a curve

Alignments on rail and road are laid out knowing that a curve entered while still slowing asks for two accelerations at right angles rather than one, so the total is what the design has to respect. This lab gives that total in one reading. The constant radius here is the simplification to watch, since a real transition changes its radius as it goes.

Ramping the feed rate along a curved tool path

A cutter or an end effector that speeds up along an arc has exactly this pair of accelerations, and the limit that matters to the machine is the total rather than either component alone. Set the radius to the arc, the speed to the feed rate and the tangential slider to the ramp, and the Total acceleration card is the figure to compare against what the machine allows.

Quoting a direction and not only a size

Saying that the acceleration on a bend points at the centre is only true while the speed holds still. The Angle off the radius card is how far that claim is out: 38.7 deg while braking, 18.6 deg powering out, and 90.0 deg from a standing start. Quote the angle beside the size whenever the speed is changing.

Working the same numbers backwards

Dragging a slider answers the forward question. If what you have is a total acceleration and you want the speed, the radius or the size of the speed change behind it, that is the same relation rearranged, and the full account of tangential acceleration works those cases through with problems and diagrams.

Where to go next

For the quantity in full, with worked problems, the formula rearranged and the diagrams that go with them, read Tangential Acceleration: What a = v2/r Leaves Out When the Speed Changes. If you would rather type figures than drag them, the tangential acceleration calculator solves the same relation for the total, the tangential term, the speed or the radius.

The steady-speed world this lab sits just outside belongs to Circular Motion Physics: Formula, Examples and Uses, which carries the definition, the symbol table and the three equivalent forms, with the circular motion lab and the circular motion calculator beside it. What Is Centripetal Force? separates the force from the acceleration, and the centripetal force lab and its calculator take the next step.

Nearby, Angular Velocity: Formula, Units and Examples covers the two rotation cells with the angular velocity lab and its calculator, the banked curve lab and the banked curve calculator handle a tilted bend, and Acceleration in Physics with the acceleration lab and the acceleration calculator covers a change of speed on a straight line. The rest is in the library of physics simulations and on the blog, and the site search will find a topic by name.