Relative Velocity Simulator

What Is the Relative Velocity Simulator?

How to use the relative velocity simulator

1. Read the four cards before you touch anything. They are already answering for two vehicles heading the same way: Relative speed  v_AB 5.560 m/s, Relative direction  bearing 90.00°, Closing speed  on the line of sight +5.560 m/s and Separation  A to B 1,000 m. The first card’s sub-line spells out what it is reporting: the magnitude of v_AB.
2. Note which letter is being watched and which is watching. The lab reads v_AB = v_A - v_B as the velocity of A as measured by B, and it says so in the paragraph above the canvas, in the fixed line below it and in the reciprocal cell. Swapping the two letters flips every component’s sign, so this is not a detail. The relative velocity guide is where the convention is argued out rather than merely declared.
3. Now move one slider, and only one: Bearing of B, from 90° to 270°. Neither speed has changed. The first card goes from 5.560 m/s to 50.00 m/s and the The same in km/h cell from 20.02 km/h to 180.0 km/h, while Difference of speeds sits there reading 5.560 m/s in both states. That cell is the whole misconception, printed beside the answer that contradicts it.
4. Read the two signed cells, v_AB east and v_AB north. They are the subtraction done one component at a time, which is the only operation that respects a direction. Press Rain and cyclist and they read -6.000 m/s and -8.000 m/s, giving a magnitude of 10.00 m/s from a three-four-five triangle. The guide to scalar and vector quantities is where that insistence on components comes from.
5. Drag Separation, B east of A and watch which cards care. The first two do not move at all; the fourth follows the slider exactly, from 0 m to 2,000 m; and the third changes because the gap it is measuring has a different direction. Take it all the way to 0 m and the Closing speed card prints an em dash instead of a figure, because two bodies at one point have no line of sight between them.
6. Press the three frame buttons in turn and watch nothing happen. Ground frame, A’s frame, B’s frame: all four cards, both component cells, the difference of speeds, the kilometres per hour and the reciprocal view are identical in every one. Only the Frame cell, the first sentence under the canvas and the arrows in the drawing change. That is the single most instructive thing on the page, and it is easiest to see on the Parallel tracks preset.
7. Read the first sentence cell under the canvas, not just the cards. It restates what each body is doing in the frame you have chosen and then gives the frame-independent answer: in B’s frame on Parallel tracks it reads In this frame A runs at 50.00 m/s, bearing 0.000° and B is at rest. v_AB is 50.00 m/s, bearing 0.000°, the same in all three frames. It is the only place the panel spells out the difference between what you see and what is true of the pair.
8. Read the second sentence cell, which tells you which case you are in. There are four of them and the slider positions decide. On Parallel tracks it opens The relative speed is 50.00 m/s, yet the separation changes at 0.000 m/s: v_AB is perpendicular to the line of sight, so none of it closes the gap. On the opening state it instead reports that the two coincide, and adds why they get muddled.
9. Watch the map rather than the panel for one run. Both bodies carry an arrow for the displacement they make over the run named in the caption, with four ticks along it, so an arrow’s length is that body’s speed in the frame now selected. A fainter pair of dots shows the same two bodies partway through, joined by their own faint line, so the gap can be watched to shrink, grow or hold rather than only read off the third card.
10. Press Pause motion when you want to look rather than watch. It stops the single animation frame and freezes the faint pair where it had got to, and no reading on the panel changes either way. The button then reads Play motion. Reset puts you back on the opening setting, in the ground frame, with the motion running.

Dragging is the right way to find out which reading a control is allowed to move, and a poor way to hit an exact figure — the speed sliders step in hundredths of a metre per second and the bearings in whole degrees. When you already have two velocities and want the answer typed out, the relative velocity calculator takes the same two speeds and bearings with kilometre-per-hour and mile-per-hour menus on both speed boxes. It has no separation and no closing speed, which are the two readings this panel adds.

Worked example: change one thing at a time

Nothing in the tables below was worked out here. Each row is a setting the lab was put into and then read back from, cell by cell, so every string in them is the panel’s own. The first table moves as few sliders as possible between consecutive rows; the second holds every slider still and presses the frame buttons instead.

Rows one and two are the pair the whole page turns on. One slider moved, through 180°, and the relative speed goes from 5.560 m/s to 50.00 m/s. The difference of the two speeds is 5.560 m/s in both rows, because a difference of speeds cannot tell a head-on pass from an overtake. Read those two columns against each other and the headline misconception is dead on the page.

Row seven is the same trap from the other side. Two bodies at 10 m/s each, one due north and one due east: the difference of their speeds is 0.000 m/s and their relative speed is 14.14 m/s. Equal speeds do not mean no relative motion; equal velocities do, which is row eight.

Row three and row eight both read zero in a column, and they mean different things. Row three has the separation slider at 0 m, so the Closing speed card is suppressed to an em dash: the quantity is undefined, not small. Row eight has the two bodies travelling together, so the Relative direction card is suppressed instead, while the closing speed is a genuine 0.000 m/s. A dash and a zero are not the same claim.

Row five is where the page earns its keep. Two aircraft on parallel tracks a kilometre apart have a relative speed of 50.00 m/s and a closing speed of 0.000 m/s, at the same time, in the same row. The rate at which that gap changes is zero, because none of the relative velocity points along the line joining them — the v_AB north cell carries all +50.00 m/s of it and the east cell reads 0.000 m/s. It is an instantaneous rate, not a promise about the next minute: let the run play out and the line of sight swings round, so the gap does creep open.

Rows ten and eleven are the same two settings with the bearings exchanged, and they separate the sign from the magnitude. Both read 500.0 m/s on the first card, which is the fastest the sliders go, and 1800 km/h on the conversion cell. The closing speeds are +500.0 and -500.0 m/s: one pair is converging on that gap and the other is tearing away from it, and only the third card knows which.

The same motion, three frames, nothing moved

Every slider below is held at the Parallel tracks setting and only the frame button is pressed. The top two rows change and the eight below them do not, which is the entire argument of this lab in one table.

Look at what the second row does to the first four speeds it names. In the ground frame A runs at 200.0 m/s and B at 150.0 m/s; in A’s frame A is simply at rest and B runs at 50.00 m/s on bearing 180.0°. Three different descriptions of how fast things are going, and one unchanged answer to how fast each sees the other.

The drawing changes more than the panel does. In A’s frame A has no arrow at all and is labelled A at rest; in B’s frame it is B that stands still and A that slides. The dashed line between them is the same length in all three, because the lab fits one metres-per-pixel scale to the pair and refuses to rescale when the frame changes.

Formula and symbol reference

The lab prints its relations on itself, in the fixed line under the canvas. Written out in the plain-digit form the panel uses, with every subscript as an underscore:

v_AB = v_A - v_B, component by component, read as the velocity of A as measured by B. Then the relative speed is the magnitude of v_AB; the relative direction is its compass bearing; the closing speed is its projection on the line of sight; and v_BA = -v_AB is equal and opposite. No modulus bars appear anywhere on this page, because a bar either side of a symbol draws as a capital letter I in the site’s fonts.

The quantity that never appears is an acceleration. There is no slider for it, no card reporting it and no curve: both bodies travel at constant velocity for the whole run, which is what lets a single arrow stand for a whole displacement. The guide to acceleration is where that restriction is lifted, and the vector addition calculator is the place to combine two velocities rather than subtract them — though it measures its angles anticlockwise from the x-axis, which exchanges the sine for the cosine.

Twelve rows follow and only five of them are something you drag. The rest are cards, cells or quantities the panel names without giving you a control for. For the sliders the right-hand column gives the end stops and the step; for everything else it gives the strings the lab prints and the span it prints them over.

If you read one row twice, make it Difference of speeds. It is the only cell on the panel that is there to be wrong in most states, and the second sentence under the canvas says in words whether the current setting is one of the few where it is right.

The physics: why changing frames changes nothing physical

Changing frame means subtracting one and the same velocity from both bodies. Whatever that velocity is, it cancels out of the difference between them, so v_AB survives the operation untouched. The three frame buttons are therefore not three answers but one answer drawn three ways, and the four cards are the part of the picture that cannot tell which button you pressed.

That is why the panel can promise an invariance rather than merely exhibit one. Pick any of the twelve settings in the table above, press all three buttons, and the relative speed, the bearing, the closing speed and the separation come back byte for byte the same. The guide to velocity and speed is where the distinction the first two cards rest on is set out: a speed has survived the subtraction as a magnitude, and a velocity has carried its direction through it.

The frame of A is the most useful of the three and the least often drawn. In it A does not move at all, and B sweeps past at the reciprocal relative velocity — on Parallel tracks, 50.00 m/s on bearing 180.0°, due south, which is the answer to “what does the pilot of A actually see out of the window?”. Both bodies are flying north over the ground, and from A the other aircraft appears to be going backwards.

The closing speed is the one reading that needs the positions as well as the velocities. It is the component of v_AB along the line joining the two bodies, so the same pair of velocities gives a different closing speed depending on where they happen to be. That is why it is the only card the separation slider can move, and why it can read 0.000 m/s while the relative speed reads 50.00 m/s.

Where the relative velocity readings break down

Every slider position this lab can reach is answered, and the arithmetic is not where the panel can let you down. The model behind it is. Six limits sit under every reading on this page, and two of them are refusals the panel makes on purpose rather than failures.

A difference of speeds is not a relative speed, and the panel prints both so you cannot forget it

This is the one misconception the whole lab is arranged around. The two agree only when both motions lie along a single line in the same sense; everywhere else they part company, and they can part company completely. The table above has a difference of 5.560 m/s beside a relative speed of 50.00 m/s, and a difference of 0.000 m/s beside 14.14 m/s. The second sentence cell states which of the two cases the current setting is in rather than leaving you to guess.

A relative speed is not a closing speed either, and the difference is not small

The third card is the projection of the relative velocity on the line of sight, so it depends on where the bodies are. On Parallel tracks the relative speed is 50.00 m/s and the gap changes at 0.000 m/s; put the same two velocities in line and the two readings coincide. A projection can never exceed what it is taken from, so the third card is never larger than the first — which is exactly why mistaking one for the other feels safe and is not.

At zero separation the closing speed is undefined, and the card says so instead of printing a number

Drag the fifth slider to 0 m and the Closing speed card prints an em dash. There is nothing wrong with the relative velocity at that setting — the first two cards carry on reading 5.560 m/s and 90.00° — but the line of sight between a point and itself has no direction, so there is nothing to project onto. Dividing by that zero would print a confident infinity, and the panel refuses rather than divides.

A zero relative velocity has no direction, so the second card is suppressed rather than computed

Press Travelling together and the Relative direction card shows a dash. Zero is a perfectly good answer on the first card — 0.000 m/s, the two travel together and neither sees the other move — but a vector of zero length points nowhere. The naive arctangent of zero over zero returns zero, which would print as a confident bearing of due north, and that single wrong readout is the reason the suppression exists.

Four significant figures is the ceiling, and it is a choice rather than a limit of the data

Every input here is an exact value you chose on a slider, so there is no measurement uncertainty to propagate and nothing in the physics decides how many digits to print. The lab stops at four, with the trailing zeros kept, which is why a reading is 50.00 m/s rather than 50 and a bearing is 90.00° rather than 90. Rounding happens once, so re-deriving one printed figure from another will not always reproduce the fourth digit.

At speeds approaching that of light the subtraction stops being the right operation

Nothing on this panel goes near that limit: 500.0 m/s is the fastest relative speed the sliders reach, and there the plain subtraction is not approximately right but right to twelve digits. Push the same rule to two craft closing at eight tenths of the speed of light and it returns 1.6 times that speed, which is not a velocity at all. The crossover for a four-figure answer sits at roughly seven thousandths of the speed of light, and what replaces the rule past it belongs to the special relativity guide.

Where relative velocity is actually used

Air traffic separation, which is a closing-speed problem and not a relative-speed one

The Parallel tracks preset is the case that makes the distinction worth money: two aircraft with 50.00 m/s between them whose 1,000 m gap is, at the instant the cards are read, changing at no rate at all. A controller cares about the third card, not the first. Press the fourth preset and you are sitting in B, watching A slide north at 50.00 m/s while the range to it barely moves — a zero closing speed buys you this moment, not the whole run.

Collision-avoidance radar, which reads the projection directly

A radar return gives the rate at which a range is changing, which is the Closing speed card and not the Relative speed card. The sign is the whole message: + closes, - opens, as the card’s own sub-line puts it. Set the two bodies head-on and the card reads +500.0 m/s at the slider tops; exchange the two bearings and the same pair of speeds reads -500.0 m/s instead.

Overtaking on a road or a railway, where the difference of speeds really is the answer

Set A to 33.33 m/s and B to 25 m/s, both on bearing 90°, 400 m apart, and the panel reads 8.330 m/s on all three of the first card, the closing speed and the difference of speeds. This is the collinear same-sense case, the one most people picture, and it is the reason the misconception survives. The The same in km/h cell reads 29.99 km/h, which is the figure a driver would recognise.

Knowing which way to tilt an umbrella

Press Rain and cyclist: rain falling due south at 8 m/s, a cyclist heading due east at 6 m/s, and the rain arrives at 10.00 m/s on bearing 216.9° — from in front and to the side, which is why you tilt forward. That same geometry is 36.87° from the downward vertical, which is the figure an umbrella angle is usually quoted as; it is the same triangle measured from a different reference, and it is not the bearing the card prints.

Collisions, where the relative velocity of approach is the input

What a collision does depends on the velocity of approach rather than on either body’s speed over the ground, which is why two cars at 27.78 and 22.22 m/s meeting head-on is a 50.00 m/s event and the same pair overtaking is a 5.560 m/s one. The panel gives you the approach velocity and stops there; the guide to elastic and inelastic collisions is where the masses and the energy come in.

Anything where a wave is involved, because the shift depends on the relative motion

A source and an observer in relative motion shift the frequency between them, and it is the closing component that does it rather than either speed alone. On Parallel tracks the closing speed is 0.000 m/s, so there would be no first-order shift at all between two bodies moving at 200 and 150 m/s. The Doppler effect guide is where that is worked out properly.

Boats and currents, which this panel deliberately cannot set up

A river crossing is a relative-velocity problem with a width and a bank attached, and neither of those is a control here: there is no boundary on the map and nothing to arrive at. The subtraction is the same operation, and the crossing time, the heading correction and the downstream drift are not on this page at all. The relative velocity guide works that case through with the numbers.

Where to go next

A panel can show you which readings a control is allowed to move, and that is the end of what a panel is for. Relative Velocity Explained is where the subject itself is written out: the convention argued rather than declared, the river crossing and the crosswind worked through, the eight problems and the misconceptions. The calculator that takes these figures typed heads the list of related tools below.

Two neighbouring ideas have pages of their own and are not argued here. The guide to velocity and speed is the distinction the first two cards rest on, and the guide to scalar and vector quantities is where the insistence on components comes from.

Further out, the special relativity guide is where the plain subtraction finally fails and the time dilation guide is one of the things that replaces it, the Doppler effect guide is a closing speed turned into a frequency, and the guide to acceleration is what happens when the velocities stop being constant.

Nearer home, the vector addition lab draws two vectors head to tail where this one subtracts them, the velocity and speed lab separates the magnitude from the direction, and the Doppler effect lab puts relative motion between a source and an observer. Beyond that there is the simulation library to work through, an index of calculators for when typing beats dragging, and the blog for everything written out at length.