Relative Motion

Relative position and velocity

Relative motion compares positions measured in two reference frames. For particle $P$, frame $A$, and frame $B$,

Differentiation gives the Galilean velocity transformation,

Subscripts read “of the first object relative to the second.” If frame $A$ moves at constant velocity relative to $B$, acceleration has the same value in both frames:

Intercept tests and closest approach.

Under constant relative velocity, the intercept condition

has a physical solution only if both component equations produce the same nonnegative time. Dividing one component equation by another is unsafe when a component of relative velocity is zero. A more robust geometric test uses the two-dimensional scalar cross product: ${\vec{r}}_0\times {\vec{v}}_{\mathrm{rel}}=0$ is required for the two vectors to be parallel. The dot product must also be negative for a future intercept, since the relative velocity must have a component toward the origin.

When this parallel condition fails, closest approach still has a definite time. The squared separation is

Its derivative is zero at

The closest-approach time is relevant only when nonnegative. If it is negative, the objects are already separating and their closest approach over the future interval is their current separation. The formula works without inventing a pursuit force or assuming either object turns toward the other.

Components in a common basis.

Vector addition requires a single coordinate basis. With east as $x$ and north as $y$, a boat's ground velocity is

Speeds add directly only for collinear vectors. A current or wind changes the ground track through vector components, so a travel time and a landing position require separate calculations.

Moving media and frame transformations

Aircraft navigation names three velocities:

• Airspeed — velocity of the aircraft relative to the air.
• Groundspeed — velocity of the aircraft relative to the ground.
• Wind — velocity of the air relative to the ground.

They combine by the frame chain

A prescribed ground track fixes the direction of ${\vec{v}}_{A/G}$, not the heading direction of ${\vec{v}}_{A/\mathrm{air}}$. A crosswind requires a heading offset whose lateral component cancels the drift. The velocity triangle fixes the required airspeed direction before the ground-track components are evaluated.

Galilean frame transformation.

Frames moving at constant relative velocity use the coordinate transformation

The same event has different position and velocity coordinates in the two frames, but acceleration is identical. This is the Newtonian form of the relativity principle for inertial frames.

The constant vertical separation in velocity is the frame-speed difference; equal slopes give the common acceleration.

Pursuit, interception, and range rate

For two particles,

With constant relative velocity, interception requires

The $x$ and $y$ components must produce the same positive time.

Rotating-frame boundary.

The Galilean transformation applies only between frames with constant relative velocity and parallel fixed axes. A rotating platform has axes whose directions change with time. Velocities measured there include additional rotation-dependent terms, so the simple subtraction ${\vec{v}}^{\prime }=\vec{v}-\vec{V}$ is incomplete.

Range rate.

Radar and navigation systems often measure range $R=|{\vec{r}}_{A/B}|$ and its time derivative rather than full relative velocity. Differentiating the magnitude gives radial range rate,

Only the component of relative velocity along the line of sight changes range. A transverse relative velocity can be large while instantaneous range rate is zero.

Relative acceleration and time histories

Relative acceleration follows from component subtraction:

It describes curvature of one trajectory as viewed from the other. Two vehicles with equal acceleration have zero relative acceleration even while their individual velocities change. A frame attached to an accelerating vehicle is not inertial, although its relative-position coordinates can still be used with the appropriate inertial-force correction.

Component trace.

Each component calculation carries its frame label.

The direct-crossing geometry and component construction represent the same frame-labelled sum.

The component diagram resolves the eastward cancellation shown by the river track.

Frame labels and closure checks.

Relative-motion notation prevents a common category error: subtracting a position from a velocity or combining two velocities that refer to incompatible pairs of objects. The ordered label $P/A$ means “particle P measured from frame A.” In the sum ${\vec{v}}_{P/B}={\vec{v}}_{P/A}+{\vec{v}}_{A/B}$, the intermediate label A closes: the first arrow ends at A and the second begins at A. The result connects P to B. This is the same cancellation rule used when adding directed displacements.

Reversing a relation changes its sign. Thus ${\vec{v}}_{A/B}=-{\vec{v}}_{B/A}$, and a stationary passenger in a train has ${\vec{v}}_{P/T}=\vec{0}$ even though ${\vec{v}}_{P/G}={\vec{v}}_{T/G}$ in the ground frame. Zero velocity is never complete without the reference object or frame. A boat “at rest” relative to water can drift rapidly relative to shore; a spacecraft “at rest” relative to a docking port can move quickly relative to a planet.

The same physical event is described at one shared Newtonian time in all Galilean frames. Positions at that event differ because their origins differ. Comparing the position of a train passenger at one ground-clock reading with the position of that same passenger at a different train-clock reading is not a transformation; it is a comparison of two different events.

Galilean transformations in components.

Let frame $S^{\prime }$ move with constant velocity $\vec{V}=(V_x,V_y)$ relative to S, with parallel axes and coincident origins at $t=0$. The transformation is applied component by component:

The inverse transformation adds the same frame velocity. This symmetry provides a quick sign check: transforming from ground to train subtracts train-over-ground; transforming back from train to ground adds it. A moving observer can describe an object as moving west while a ground observer describes it as moving east. Both use the same acceleration when their frames have constant relative velocity.

Relative position and separation histories.

The relative-position vector ${\vec{r}}_{A/B}={\vec{r}}_A-{\vec{r}}_B$ points from B to A. Its magnitude is separation, but the vector also retains bearing. A constant relative velocity produces a straight relative-position trajectory. The origin of this relative coordinate system is attached to B, so B remains fixed at zero and A appears to move with ${\vec{v}}_{A/B}$. This change of description can turn two moving ground-frame trajectories into one moving point and one fixed origin.

Separation alone is not enough to determine an intercept. The range may decrease at first and then increase if the relative-velocity vector does not point exactly toward the origin. A genuine intercept requires the full vector equation ${\vec{r}}_{A/B}(t)=\vec{0}$. In two dimensions, the starting separation and relative velocity must be parallel and oppositely directed for a constant-velocity collision to occur. A nonzero sideways component produces a closest approach rather than a meeting.

Fixed-speed pursuit and feasibility.

An interceptor that can select its heading but has a fixed speed relative to a medium faces a different problem from passive relative motion. Let the target have known ground velocity ${\vec{v}}_T$, and let the interceptor have fixed ground-speed magnitude $u$. An intercept at time $t$ requires its constant ground velocity to be

The heading is feasible only if $|{\vec{v}}_I|=u$. Solving that magnitude condition for positive $t$ finds possible intercept times. A solution can fail to exist when the interceptor is too slow in the relevant direction. Being faster in speed alone is not always enough when a target begins far ahead and the allowed time or region is restricted.

The geometry can be viewed in velocity space. For a selected time, the displacement needed to reach the target fixes a required average ground velocity. As time grows, the displacement-per-time contribution shrinks and the required velocity approaches the target velocity. Very short intercept times demand very large speed; very long times may be physically possible but tactically irrelevant. The mathematical root must be checked against any specified time window and against the assumption that both velocities remain constant.

Relative acceleration and noninertial observers.

Relative acceleration is always obtained by subtracting accelerations measured in a common inertial frame: ${\vec{a}}_{A/B}={\vec{a}}_A-{\vec{a}}_B$. Equal accelerations give zero relative acceleration, so separation can change linearly even while both objects accelerate in the ground frame. The same conclusion covers two cars braking equally and two objects dropped in the same local gravitational field.

An observer attached to one of those accelerating objects is not an inertial frame. The relative-coordinate equations still describe geometry, but Newton's second law in that accelerating observer's frame requires an additional inertial-force term. The ordinary Galilean statement that all observers agree on acceleration applies only to frames moving at constant relative velocity. A rotating observer adds still more terms because its axes change direction as well as its origin's motion.

Range, bearing, and collision assessment.

Range rate measures only the radial part of relative velocity. A negative range rate means the distance between two objects is decreasing at that instant; a positive range rate means it is increasing. Neither quantity alone determines collision risk. An object can have a strongly negative range rate and still pass at a safe lateral distance, while an object with zero range rate may be at closest approach before moving away. The relative-position vector and the transverse component of relative velocity supply the missing geometric information.

Bearing is the direction of the relative-position vector. In planar motion, a constant bearing together with decreasing range is a warning sign: the relative velocity is directed along the line of sight, so the relative-position vector can reach zero. A changing bearing usually signals a transverse component and hence a miss, though a changing reference orientation or noisy angle measurement must not be mistaken for physical transverse motion. Navigation systems often use both range and bearing histories because neither record alone identifies the complete relative velocity vector.

The radial-transverse split is found by projecting relative velocity onto the unit line-of-sight vector. The radial projection changes range; the remainder changes bearing. At closest approach the radial projection is zero, so relative velocity is perpendicular to the separation vector. This perpendicular condition is the geometric version of setting the derivative of squared range to zero.

Bearing-based interpretation.

A ship observes another vessel at a bearing that remains fixed while range falls from $4.0\mathrm{km}$ to $2.0\mathrm{km}$ over ten minutes. The average radial range rate is $-0.20\mathrm{km}\mathrm{mi}{\mathrm{n}}^{-1}$. Under the stated constant-bearing and constant-relative-velocity model, the relative path lies along the line of sight and collision would follow ten minutes later if neither vessel changes course. If the bearing changes by several degrees over the same interval, the conclusion is no longer valid without a full relative-vector calculation.

Solution methods and coordinate changes

Start by naming the objects and selecting one inertial frame as the common basis. Write every supplied velocity with its two labels, then translate all vectors into components in that same basis. A diagram may suggest the direction of a result, but component equations determine its signs and allow a numerical check. Keep speed separate from velocity: a speed is a magnitude and cannot be inserted into a vector sum until a direction or component decomposition is supplied.

For crossing and wind problems, identify which velocity is constrained by the question. A specified heading constrains the vehicle-medium velocity; a specified ground track constrains the vehicle-ground velocity. For interception, form relative position and relative velocity before solving time. For a frame conversion, state the frame velocity and use subtraction in the requested direction. These choices prevent an otherwise correct triangle from answering a different physical question.

After solving, test limiting cases. With zero current or wind, the ground and medium velocities should coincide. With two objects sharing the same velocity, relative velocity should vanish. With equal accelerations, relative acceleration should vanish. A proposed direct river crossing becomes impossible if current speed exceeds the available through-water speed, because no heading can supply a lateral component large enough to cancel the current. Such checks expose sign mistakes and impossible geometric assumptions before numerical rounding hides them.

Reporting an intercept result.

An intercept answer needs more than a time. A complete statement identifies the reference frame, the intercept location or required heading when relevant, and the model interval over which velocities were assumed constant. “Interception occurs after $45\mathrm{s}$” is incomplete if the target is moving through a current or if the pursuer has a finite speed constraint. “In the ground frame, the craft reaches the target after $45\mathrm{s}$ by holding the stated heading” makes the relation between the calculation and the physical claim reproducible.

Common failure modes.

Three mistakes recur in relative-motion work:

• Adding speeds, not vectors — combining two speeds rather than two labeled velocity vectors loses direction and frame information.
• Wrong reference medium — using a ground speed where the problem supplied a speed relative to air or water changes the physical triangle before any calculation begins.
• Accepting a negative time — a negative intercept time ignores the direction of relative motion; it usually places the intersection in the past.

A brief label audit and a sign check on the final time prevent all three.

Relative coordinates also clarify apparently paradoxical statements. Two objects can move rapidly in a ground frame while remaining fixed relative to each other. Conversely, an object stationary in one frame can move in every other frame whose origin has nonzero relative velocity. The frame label resolves the apparent contradiction without changing any physical event.

Relative acceleration as a vector time history.

Relative acceleration describes how one object's velocity changes as seen from the other. In a common inertial frame,

The equation remains valid when either acceleration changes with time. Integrating it over an interval gives the change in relative velocity, and integrating again gives the change in relative position. Initial relative position and initial relative velocity are still required: acceleration history alone determines neither the initial separation nor the current closing speed.

Constant relative acceleration gives the relative trajectory the same form as ordinary constant-acceleration motion,

It gives the difference between two independently measured motions. A chaser can have a larger ground-frame acceleration than a target but still move away initially if its relative velocity points outward. The acceleration controls curvature of the relative path; the initial relative velocity controls its initial tangent.

When acceleration is supplied as a sampled vector history, componentwise updates are safer than using changing magnitudes. Over a short interval, add ${\vec{a}}_{A/B}\Delta t$ to relative velocity, then use the appropriate average relative velocity to update relative position. A changing direction of acceleration can rotate the relative velocity even when its magnitude is nearly constant. A single scalar “closing acceleration” is adequate only when all relevant vectors are collinear with the line of sight.

Changing origins without changing the events.

A frame transformation has two parts: a translation of the origin and, possibly, a relative velocity of the origins. If the primed origin has ground-frame position $\vec{R}(t)$, then coordinates of the same event obey

Differentiation gives ${\vec{v}}^{\prime }=\vec{v}-\dot{\vec{R}}$ and ${\vec{a}}^{\prime }=\vec{a}-\ddot{\vec{R}}$. The familiar Galilean form is the special case $\vec{R}(t)={\vec{R}}_0+\vec{V}t$, for which $\ddot{\vec{R}}=0$. Writing the origin trajectory explicitly makes clear why a frame attached to an accelerating car does not share the ground-frame acceleration values.

At constant frame velocity, the subtracted displacement is $\vec{V}t$ over the same elapsed time.

An origin shift at one fixed time changes every displayed position by the same vector. It does not alter separations: for two objects A and B,

The cancellation leaves relative position independent of the arbitrary coordinate origin. Two observers can therefore disagree about each object's map coordinate while agreeing exactly on the displacement from B to A. The same cancellation holds for a common constant-velocity frame translation in relative velocity, but fails if one observer mistakenly subtracts a velocity measured in a different direction or at a different time.

The same origin shift applies to positions recorded at each common clock time; it does not pair positions from different events.

Vector time histories and integration order.

Velocity and acceleration histories are vector functions, not sequences of unrelated arrows. A velocity arrow at a later time begins at the same velocity-space origin as an earlier arrow; its difference is the accumulated acceleration area. A position arrow at a later time differs from an earlier position arrow by the accumulated velocity area. Drawing these histories tail-to-tail prevents the common mistake of connecting successive velocity tips as though they were locations in physical space.

A piecewise-constant relative-acceleration table should record the interval, relative acceleration, beginning relative velocity, ending relative velocity, and relative displacement. The ending values carry forward. This procedure also records when an assumed model changes: a turn, engine burn, or control input belongs at an interval boundary rather than being smeared across the entire history.

Measurement, uncertainty, and feasibility

Closest approach is a relative-motion calculation with a geometric condition rather than an assumption that the objects meet. Place object B at the origin of the relative coordinate system at the initial time. The position of A is then ${\vec{r}}_0$, and its straight-line relative path is

At the minimum separation, the vector from B to A is perpendicular to the relative-velocity vector. Otherwise a component of velocity would still be reducing or increasing the separation. The condition is

which yields the closest-approach time already obtained by differentiating squared range. Squared range is preferred over range itself because it avoids a square root and retains the same minimum for nonnegative distance.

The calculation must distinguish three outcomes. A positive closest-approach time and zero closest distance gives an intercept. A positive time and nonzero distance gives a future miss. A negative closest-approach time means the mathematical minimum occurred before the selected initial time; from now on, the objects separate under the constant-velocity model. These statements are about the relative path, not about which object is considered the pursuer.

Model limits in encounter prediction.

Closest-approach predictions assume that both ground velocities remain constant over the computed interval. A turn, a wind change, a control response, or a different altitude invalidates the straight relative path after the change. The calculation is still applicable as a short-horizon prediction, but its conclusion should be reported with the time window and the velocity source. It predicts geometric proximity, not intent, communication, or collision avoidance behavior.

Sensitivity and uncertainty in relative predictions.

Relative calculations combine measurements from two objects, so their uncertainty often exceeds the uncertainty of either individual measurement. A small error in each velocity component changes the relative velocity by their difference. Over a long prediction time, that velocity error becomes a growing relative-position error. For constant relative velocity, an uncertainty $\delta \vec{v}$ contributes roughly $t\delta \vec{v}$ to the predicted separation after time $t$. A reported intercept time should therefore not carry more precision than the position and velocity measurements allow.

The closest-approach time is especially sensitive when the dot product ${\vec{r}}_0\cdot {\vec{v}}_{\mathrm{rel}}$ is small. In that situation the objects are already close to perpendicular relative motion, and a small change in a measured velocity component can shift the predicted closest time from slightly future to slightly past. The minimum separation itself can be more robust than the time, or the reverse, depending on the geometry. Reporting both the input time interval and the measurement precision gives the calculation its proper scope.

Sensitivity can be recorded by separating the inputs that perturb event time from those that perturb miss distance. The same speed uncertainty does not affect both outputs equally for every encounter geometry.

Relative velocity can also be inferred from repeated range-and-bearing observations, but numerical differentiation amplifies angle and range noise. Fitting a straight relative trajectory to several observations is often more reliable than subtracting two nearly equal position readings from a single pair of frames. The fitted model should still be checked against residuals: systematic curvature indicates turning, changing wind, or a noninertial reference issue rather than random measurement scatter.

Assumption audit.

Every relative-motion result rests on an observation time, a coordinate convention, and a model for how velocities evolve afterward. Verify that both velocity vectors were measured in the same frame, their timestamps were synchronized, and constant velocity or constant relative acceleration remained credible over the forecast interval. A numerical answer can be algebraically correct and physically unusable when any of those conditions is missing. Naming the reference frame and forecast horizon turns a vector result into a testable statement about a real encounter.

The encounter record should expose the calculation in the same order as the physical inference. A scalar range or speed is insufficient when the transverse component controls the miss geometry.

Derivatives of relative coordinates.

Relative coordinates form an ordinary vector-valued function of time. Once ${\vec{r}}_{A/B}(t)={\vec{r}}_A(t)-{\vec{r}}_B(t)$ has been defined at common times, differentiation produces relative velocity and relative acceleration without any new kinematic rule. The important condition is synchronization: positions from different time stamps cannot be subtracted to estimate an instantaneous separation, and velocities taken from different instants cannot be subtracted to estimate an instantaneous relative velocity.

Repeated differentiation clarifies what each graph or sensor record can support. Position differences yield relative position. Their time derivative yields relative velocity. A further derivative yields relative acceleration. Each derivative removes one constant of integration, so the reverse process needs an initial condition at the appropriate level. Starting from relative acceleration and integrating twice without an initial separation and relative velocity produces a family of possible encounter paths, not one prediction.

The magnitude of relative position, $R=|\vec{r}|$, has a derivative that depends on direction as well as magnitude. The range rate is the projection of relative velocity onto the current line of sight. A large transverse velocity leaves range unchanged at one instant but rotates the line of sight. In planar motion, the rate of change of bearing is proportional to the transverse component divided by range. Thus the same sideways speed produces a faster bearing sweep when the objects are near than when they are far apart.

Tracking requires this derivative structure. A radar that reports range but not bearing cannot recover a full Cartesian relative-velocity vector from one instant. Conversely, a camera that reports bearing but no range cannot distinguish a nearby slow target from a distant fast one using a single frame. Multiple time samples and a stated motion model turn these partial observations into a relative trajectory.

Feasibility before solving an intercept time.

An algebraic intercept time is meaningful only after the geometry and motion constraints have been checked. For two constant-velocity objects, the relative position must be driven to zero by a single scalar time. In two dimensions, both component equations must agree. Different east and north component solutions describe different hypothetical meeting events. No common intercept time exists.

Fixed-speed interception adds a second constraint. The required interceptor velocity must have the allowed magnitude and must satisfy any heading, acceleration, or operating-region limits. A boat cannot exceed its through-water speed, and an aircraft cannot turn instantaneously under the stated model. Such a solution is infeasible under the constraints; report that outcome instead of a rounded time with an impossible velocity attached.

The sign of time is a separate feasibility test. A negative root identifies an intersection of the straight paths before the chosen initial observation. It may be appropriate for reconstructing a past encounter, but it does not predict a future intercept. A zero root means the objects are already coincident at the selected initial time. Positive roots should still be checked against the interval over which the velocities are credible.

When relative acceleration is present, the relative-position equation can yield more than one positive root. The objects may meet, separate, and meet again after a turn or reversal. Each root must be tested against the piecewise motion stage that produced it. A quadratic solver cannot decide whether a motor was still running or a braking stage had already ended.

Radar observations and Cartesian relative data.

Radar observations are naturally polar: range $R$ and bearing $\theta$. Cartesian relative coordinates are obtained only after a direction convention is stated. With east as the positive horizontal axis and bearing measured counterclockwise from east,

Navigation displays often measure bearing clockwise from north instead. Using the same sine and cosine pair without converting the angle rotates or reflects the relative position. The safest method is to sketch the axes, identify the component adjacent to the stated bearing, and test a cardinal direction: a bearing due north must produce zero east component and positive north component in the chosen basis.

Successive polar observations also require care during differentiation. A changing range and a changing bearing both contribute to Cartesian relative velocity. The radial term points along the line of sight, while the angular term is perpendicular to it. A tracker that reports a constant range and rapidly changing bearing is not observing a stationary target; it is observing predominantly transverse motion. Likewise, a steady bearing with decreasing range is consistent with radial closing, but only the full vector history establishes whether the track stays exactly radial.

Cartesian conversion puts measurements from separate sensors into one component system. Once all observations use the same origin, time base, and axes, position differences and fitted velocity components can be compared directly. It also exposes a frequent mistake in hand sketches: range is a scalar length, while the Cartesian pair retains the direction needed for addition and subtraction.

Uncertainty propagation in relative predictions.

Uncertainty in relative position comes from both position measurements and from the coordinate transformation used to compare them. When independent Cartesian position components have comparable random uncertainty, subtracting two positions increases the uncertainty in each relative component by a root-sum-square combination. A small uncertainty in bearing can dominate the transverse coordinate at long range, because an angular error corresponds to a sideways distance approximately equal to range times the angle error in radians.

Velocity uncertainty grows when it is inferred from short time intervals. Dividing a position difference by a smaller interval improves temporal resolution but magnifies the same position noise in the velocity estimate. For encounter forecasting, a fitted relative trajectory through several measurements often gives a more stable velocity estimate than one pair of observations, provided the residuals support the constant-velocity model.

Prediction uncertainty expands with forecast time. A component error that is modest at the current observation can shift a predicted intercept or closest approach by a large distance later. The report should distinguish uncertainty in range, uncertainty in bearing, uncertainty in inferred velocity, and uncertainty caused by model change. Combining them into a single unexplained error bar hides which new measurement would most improve the forecast.

Measurement-priority decision.

The next measurement should target the component that limits the decision. When bearing uncertainty dominates a long-range forecast, a better angular observation is more valuable than another precise range reading. When a predicted closest approach is highly sensitive to relative speed, a longer time baseline between synchronized position observations can improve the velocity estimate. The relative-coordinate model therefore identifies the current prediction and the observation most likely to reduce its uncertainty.