Motion Graphs · study.

Derivatives and graph interpretation

Position $x (t)$ , velocity $v_{x} (t)$ , and acceleration $a_{x} (t)$ are successive time derivatives:

v_{x} = \frac{d x}{d t}, a_{x} = \frac{d v _{x}}{d t} = \frac{d ^{2} x}{d t ^{2}} .

The slope of a position-time graph is velocity; the slope of a velocity-time graph is acceleration. Positive acceleration does not necessarily mean increasing speed: speed decreases whenever velocity and acceleration have opposite signs. A horizontal position graph has zero instantaneous velocity, while a horizontal velocity graph has zero acceleration.

Integration reverses differentiation. Between times $t_{i}$ and $t_{f}$ ,

Δ x = t_{i} \int t_{f} v_{x} d t, Δ v_{x} = t_{i} \int t_{f} a_{x} d t .

The areas are signed. A region below the time axis contributes negative displacement or negative velocity change. Total distance requires integrating $∣ v_{x} ∣$ , not simply taking the signed area.

The three graph forms describe the same one-dimensional motion. Position slope gives velocity, velocity slope gives acceleration, and the shaded signed area under the velocity graph gives displacement over the selected time interval.

Reading the shape of a position graph.

A position graph records coordinate, not path length. Its height gives the object's position relative to the origin; its tangent slope gives velocity:

Upward slope — positive velocity.
Downward slope — negative velocity.
Steep versus shallow — larger speed, when both axes share a scale.
Leveling off — velocity approaching zero, even far from the origin.

Curvature gives acceleration. A concave-up trace has increasing slope, hence positive acceleration; a concave-down trace has decreasing slope, hence negative acceleration. Signed velocity, not apparent steepness, fixes the direction of motion. A curve can decrease while concave up: its velocity is negative but becoming less negative, so the object slows while its acceleration is positive.

The tangent is local. A chord drawn between two widely separated position points gives average velocity over that interval, not the instantaneous velocity at either endpoint. The two coincide only for a straight position-time graph. A long chord and a tangent therefore answer different kinematic questions.

The slope of a position-time trace is signed velocity, while its curvature is signed acceleration. The descending concave-up curve has negative velocity but positive acceleration; the tangent is negative and becomes progressively less steep as the object approaches its turning point.

State which line supplied the rise and run before calling a slope instantaneous.

Velocity graphs: direction, reversal, and distance.

The sign of a velocity graph has a direct spatial meaning. Above the time axis, the coordinate increases; below it, the coordinate decreases. A crossing through zero marks a momentary rest and, when the graph actually changes side, a reversal of direction. Touching the axis and remaining on the same side does not reverse the motion: it describes a brief stop followed by travel in the original direction.

Displacement is the signed area between the velocity trace and the time axis. Distance adds the magnitudes of those regions. Thus a trip with $+ 12 m$ of signed area followed by $- 5 m$ has displacement $+ 7 m$ and distance $17 m$ . The two quantities agree only when velocity retains one sign over the interval.

The positive and negative regions beneath a velocity-time graph have opposite effects on displacement. Their absolute areas both contribute to total distance; the zero crossing identifies the direction reversal between the outward and return segments.

Acceleration graphs and changes in velocity.

Acceleration is read from a velocity graph as a slope, but its own graph is read through signed area. A constant positive acceleration appears as a horizontal line above zero. Over any selected interval, the rectangular area under that line is the positive change in velocity. A constant negative acceleration appears below zero and produces a negative velocity change. Neither sign alone says whether the object speeds up: that conclusion also requires the sign of the current velocity.

For example, a car moving in the positive direction with $a < 0$ loses speed. A cart moving in the negative direction with the same $a < 0$ gains speed, since its velocity becomes more negative. The velocity-time graph separates these cases plainly. Its slope is negative in both, while the graph lies above the axis in the first case and below it in the second.

Units follow directly from the area operation. On an acceleration-time graph, vertical units are $m s^{- 2}$ and horizontal units are seconds, so area has units $m s^{- 1}$ , the required units for a velocity change. A graph whose vertical scale is labelled merely “acceleration” without units cannot support a numerical calculation until the scale is supplied.

Piecewise graphs and boundary times.

Motion graphs can change rule when a motor starts, a brake engages, or a measured force switches level. The graph should preserve the physical quantities that remain continuous. Position is continuous for ordinary motion; an object cannot jump from one coordinate to another without passing through the intermediate positions. Velocity is also continuous unless an idealized impulse is being used. Acceleration can jump abruptly when the net force changes, and that jump appears as a corner in the velocity graph or as a step in the acceleration graph.

Piecewise calculations proceed interval by interval. The final position and velocity from one interval become the initial values for the next. Reusing the original initial velocity in the second interval erases the first stage. Each new signed acceleration area changes the velocity already reached, so the prior endpoint becomes the initial value of the next interval.

A step in acceleration creates a corner, not a break, in the velocity graph. The first and second slopes equal the two acceleration values; continuity at the switch time preserves the accumulated velocity from the first stage into the second.

Reconstructing motion from graphs

An acceleration graph does not determine a unique velocity graph by itself. Its area gives only a change in velocity, so one initial velocity value is required to set the vertical placement of the reconstructed graph. The same distinction appears one level lower: integrating velocity determines a change in position, and one initial position value sets the corresponding position graph. Derivatives remove these constants; integrals must restore them from stated conditions.

A stepwise acceleration record is reconstructed exactly with geometry. Each constant acceleration interval adds a rectangular signed area to the preceding velocity. A sloping acceleration interval produces a curved velocity trace, because the slope of the velocity graph itself is changing. The value of acceleration at a single time fixes only the local slope of velocity, not the velocity’s height above or below the axis.

Estimating areas from a plotted curve.

Many laboratory graphs do not supply a convenient formula. Area is estimated from the plotted values; the trace height alone does not supply an area. The trapezoidal rule replaces a curved segment between adjacent times by a straight chord. For equal intervals $Δ t$ , the displacement estimate is

Δ x \approx i \sum \frac{v _{i} + v _{i + 1}}{2} Δ t .

Each trapezoid retains its sign. A chord below the axis gives a negative contribution; no separate minus sign is invented after the calculation. Narrower intervals usually improve the geometric approximation for a smooth velocity curve, but they do not cure a poor vertical calibration or noisy readings. A reported area should reflect the resolution of the graph, not the number of digits on a calculator.

The trapezoidal rule estimates displacement from sampled velocity by replacing each curved segment with a chord. The shaded trapezoids preserve signed area and become a better geometric approximation as the sampling intervals narrow for a smooth, well-resolved velocity trace.

Scale, units, and what a graph can justify.

Graph calculations are numerical measurements. A slope is found from coordinate differences on the axes, not from the angle a line appears to make on a page. A velocity graph can look steep because the vertical axis has been stretched; that visual steepness has no independent physical meaning. The axis scales convert the chosen rise and run into $m s^{- 1}$ or $m s^{- 2}$ as required.

Two points far apart on a straight segment generally reduce the relative effect of line thickness and reading uncertainty. They need not be plotted data points. Grid intersections on the drawn best-fit line are often better choices, provided they are actually on that line. Reading the slope from two noisy adjacent markers treats measurement scatter as if it were the physical graph.

The same discipline applies to areas. An area estimated by counting squares must include the scale represented by one square in both directions. If one horizontal square represents $0.50 s$ and one vertical square represents $0.20 m s^{- 1}$ , a square under a velocity graph represents $0.10 m$ , not a dimensionless unit. Negative regions should be counted with their sign before positive and negative totals are combined.

Worked example. A velocity line rises from $1.0$ to $5.0 m s^{- 1}$ between $2.0$ and $6.0 s$ . Its slope is $(5.0 - 1.0) / (6.0 - 2.0) = 1.0 m s^{- 2}$ . Changing the printed width of the graph changes the visual angle but cannot change this acceleration. Over the same four seconds, the trapezoid area is $[(1.0 + 5.0) /2] (4.0) = 12 m$ . Both results follow from labelled axis values, so they remain valid if the graph is redrawn at a different size.

When a value is read from a coarse graph, its precision should match the grid and line thickness. Reporting $1.0000 m s^{- 2}$ from a graph whose smallest division is $0.2 m s^{- 1}$ over one second overstates the evidence. A rounded value or a stated reading uncertainty is the more accurate scientific claim. The graph supports only the precision that its scale and data warrant.

State the graph operation before stating the physical conclusion. The following record separates a derivative, an integral, and a visual feature that does not by itself determine a cause.

Measured feature	Operation	Supported statement
Position slope	$v \approx Δ x /Δ t$	average or local velocity over the stated interval
Velocity slope	$a \approx Δ v /Δ t$	acceleration estimate with units of $m s^{- 2}$
Velocity area	$Δ x \approx \sum \overset{v}{ˉ}_{i} Δ t_{i}$	signed displacement over the selected interval
Curve shape	no numerical operation	direction change or concavity only when resolution supports it

Consistency across position, velocity, and acceleration.

The three graph types describe one motion, so a proposed set can be tested without calculating every coordinate. A horizontal velocity segment requires a straight position segment, because a constant slope in position gives constant velocity. A rising straight velocity segment demands a concave-up position curve and a horizontal positive acceleration segment. A horizontal acceleration segment does not make position horizontal; it makes the slope of the velocity graph change at a uniform rate.

The same constraints can be read in reverse. A position graph with a local maximum has zero velocity at the summit. If the graph is concave down there, the acceleration is negative, so the velocity crosses from positive to negative. The velocity graph should therefore pass through its axis with a negative slope, and the acceleration graph should lie below its axis nearby. Any alternative set of graphs violates at least one derivative relation.

One constant-negative-acceleration motion in three aligned records. Position is concave down and reaches a maximum when velocity crosses zero; the velocity-line slope equals the horizontal acceleration value. Corresponding marked times align vertically across all panels.

The common time scale is essential. Aligning graph panels only by their left edges while allowing different horizontal scales can create false correspondences. Before using a peak in $x (t)$ to locate a zero in $v (t)$ , verify that both time axes share the same origin and interval markings. A shifted clock changes the numerical time of every feature but does not alter the velocity or acceleration values.

Discontinuities and physical idealizations.

Lines and steps in textbook graphs often idealize changes. A step in acceleration is physically reasonable as a short-hand for a force changing over a time too brief to resolve. Velocity stays continuous across that step, but its slope changes abruptly. Position remains continuous as well. The resulting velocity graph has a corner, and the position graph changes curvature without developing a break.

A genuine jump in velocity is a stronger idealization. It represents an impulse whose duration is treated as zero while its velocity change remains finite. Such a vertical segment cannot be an ordinary velocity-time trajectory at finite acceleration; its slope would be infinite. In a measured collision, the transition occupies a small but nonzero time interval, and the exact rounded corner depends on the sensor’s time resolution.

A step in acceleration preserves continuous velocity, while an ideal impulsive velocity change is drawn as a vertical jump. The latter compresses a brief collision into zero plotted time and therefore cannot be assigned an ordinary finite acceleration over the jump itself.

Position jumps require a different warning. A discontinuous plotted position can occur when a tracking system loses and reacquires an object, when a coordinate origin is reset, or when two separate runs have been concatenated. It is not a kinematic path through space. A good analysis distinguishes a physical rapid change from a change in the data-recording convention before differentiating the graph.

Averages, sampling, and numerical reconstruction

An average velocity is a displacement divided by elapsed time. On a velocity-time graph it is the height of a horizontal line that would enclose the same signed area over the chosen interval. This equal-area line need not pass through the graph at the temporal midpoint. When velocity varies smoothly, it does match the velocity at some time in the interval, but graph data alone may not identify that time uniquely.

Average acceleration has the analogous meaning: it is the net velocity change divided by time. A fluctuating acceleration graph can have zero average while still altering the velocity substantially during the interval. Positive and negative acceleration areas then cancel only in the final velocity change. They do not cancel the displacement accumulated while velocity was elevated or depressed.

The interval must be named. Extending an interval can change an average even when the local graph around the original interval is unchanged. A speedometer reading is an instantaneous speed; a route distance divided by the travel time is an average speed. Instantaneous speed refers to one time; average speed refers to a named interval, despite the common unit $m s^{- 1}$ .

Numerical reconstruction with a table of samples.

Graph relations also work when the source is a numerical table. A sequence of velocity samples reconstructs position changes by summing interval areas. With measurements at uniform time spacing, the trapezoidal estimate uses the average of the two endpoint velocities in each interval. The result is a list of positions, not an assertion that the hidden velocity ran linearly between every pair of measurements.

Acceleration samples can be accumulated one layer earlier to obtain velocities, then accumulated again to obtain positions. Errors and offsets propagate through this process. A constant $+ 0.05 m s^{- 2}$ bias in acceleration creates a velocity error that grows linearly with elapsed time and a position error that grows quadratically. Long reconstructed records therefore need independent position or velocity checks, especially when acceleration comes from a sensor whose zero is hard to calibrate.

Full multi-interval graph interpretation.

A complete motion-graph calculation begins with the time boundaries and the acceleration or velocity rule on each interval. The ending velocity and position from one interval become the initial values of the next. Velocity signs identify direction, signed velocity areas give displacement, and absolute areas give distance. This order keeps local derivative information separate from accumulated integral information.

The following acceleration history has three stages: positive acceleration, zero acceleration, and negative acceleration. It starts from a negative velocity, so the first stage contains a direction reversal. The velocity graph must begin below its axis, rise linearly, remain horizontal, then fall linearly. The position graph first decreases, reaches a minimum when velocity becomes zero, then rises; during the middle stage it is a straight rising line, and during the final stage it remains concave down while its slope diminishes.

Worked example. Take the first interval as $0 \leq t < 2.0 s$ with $a = + 2.0 m s^{- 2}$ and initial velocity $v_{0} = - 2.0 m s^{- 1}$ . The velocity change is $+ 4.0 m s^{- 1}$ , so the interval ends at $+ 2.0 m s^{- 1}$ . The zero crossing occurs one second after the start, not at the interval boundary. The signed displacement over this first interval is the symmetric trapezoid area,

Δ x_{1} = \frac{- 2.0 + 2.0}{2} (2.0) = 0.

Zero displacement does not mean no travel. The negative triangle from $0$ to $1.0 s$ has area $- 1.0 m$ and the positive triangle from $1.0$ to $2.0 s$ has area $+ 1.0 m$ . The distance in the first stage is therefore $2.0 m$ , while the object returns to its starting coordinate at the stage boundary.

During the second interval, from $2.0$ to $5.0 s$ , let $a = 0$ . Velocity remains $+ 2.0 m s^{- 1}$ and the position graph is a straight line. The velocity rectangle gives

Δ x_{2} = (2.0) (3.0) = 6.0 m .

For the third interval, from $5.0$ to $7.0 s$ , let $a = - 1.0 m s^{- 2}$ . The velocity falls from $+ 2.0 m s^{- 1}$ to zero. Its triangular area is

Δ x_{3} = \frac{2.0 + 0}{2} (2.0) = 2.0 m .

The total displacement is $0 + 6.0 + 2.0 = 8.0 m$ , and total distance is $2.0 + 6.0 + 2.0 = 10.0 m$ . The final velocity is zero at a position different from the starting coordinate. Different graph features determine the three results: endpoint height gives final velocity, signed area gives displacement, and the magnitudes of all travelled segments give distance.

Audit questions for a graph-based answer.

Before finalizing a result, check four points:

the velocity slope matches the acceleration in every interval;
the velocity area reproduces the stated position change;
a direction reversal carries a velocity sign change;
reported units match the operation, slope or area.

These checks locate most sign errors without a second solution method.

What motion graphs determine

Even a complete set of one-dimensional graphs does not explain the cause of the motion. A velocity line with negative slope establishes negative acceleration; it does not by itself identify friction, gravity, a braking force, or a motor setting. Several physical systems can produce the same kinematic record. Force claims require additional information about the object and its interactions. Keeping this boundary clear prevents a graph-reading exercise from importing assumptions that were never measured.

Derivative and integral claims follow from graph data; force claims require interaction data. State that evidential boundary explicitly so a kinematic conclusion does not acquire an unmeasured mechanism.

Graph evidence	Quantitative inference	Unsupported inference
negative slope of $v (t)$	negative acceleration in the chosen axis	which force produced it
zero crossing of $v (t)$	instantaneous reversal of direction	zero net force
signed area under $v (t)$	displacement over the interval	total distance without sign analysis
curve between samples	interpolation model	unmeasured instantaneous detail

Reference frame and coordinate choice determine the graph labels and signed values. Reversing the positive axis reverses the signs of position, velocity, and acceleration, while distance and speed remain nonnegative. Translating the position origin shifts the whole position graph vertically but leaves its slopes and curvature unchanged. A frame moving at constant speed changes a velocity graph’s vertical placement but leaves the acceleration graph unchanged. These transformations alter labels and values, not the underlying observed motion.

Sampling and line thickness set further limits. A smooth curve drawn through sparse points is an interpolation, not direct evidence about every intervening instant. Reports should distinguish a measured feature from a modelled feature: “the sampled velocities support a decreasing trend” is warranted more directly than a precise acceleration history below the time resolution. State those limits alongside the calculation.

Stationary points, extrema, and inflection points.

Several graph features have related but distinct kinematic meanings. A stationary point on a position graph has zero tangent slope, so its velocity is zero at that instant. It may be a maximum, a minimum, or neither. A local maximum requires the position graph to rise before the point and fall after it; velocity changes from positive to negative. A local minimum requires the reverse sign change. A flat shoulder can have zero velocity without reversing direction when the slope touches zero and retains its sign.

An extremum on a velocity graph is different. Its slope is zero, so acceleration is zero there. The object may still be moving rapidly. A speed maximum is found from the magnitude of velocity, which means a positive velocity maximum or a negative velocity minimum can each represent a largest speed. Treating every horizontal tangent as “the object stops” confuses the graph’s vertical quantity with the derivative of that quantity.

An inflection point on a position graph marks a change in concavity. It is commonly associated with acceleration changing sign, but data with a sharp corner or coarse sampling may not support a precise statement about the instantaneous acceleration. A smooth trace has a velocity-graph local extremum at the same time as the position curve changes concavity. The three features align by time, not by their vertical graph values.

Classifying a zero-velocity instant.

At $t = 2.0 s$ , a position graph has a horizontal tangent. Immediately before and after that time, the graph rises. The velocity is zero at the marked instant but positive on both sides, so the object pauses without reversing. If the position graph is concave up at the pause, acceleration is positive and the velocity touches zero from above only in an idealized limiting sense; a smooth physical trace with positive velocity on both sides instead has a flat local slowdown whose detailed shape must be resolved by the data. A velocity sign change establishes whether a turnaround occurred; a horizontal tangent alone does not establish one.

Cumulative area as a new graph.

Fixing an initial position turns a velocity graph into a cumulative-displacement function. Define $D (t)$ as the signed area under velocity from the start time to time $t$ . Then $x (t) = x_{0} + D (t)$ . The value of $D$ rises while velocity is positive, falls while velocity is negative, and has a horizontal tangent when velocity is zero. The cumulative-area graph is the reconstructed position graph apart from the vertical offset $x_{0}$ .

The cumulative graph clarifies interval areas. The displacement from $t_{1}$ to $t_{2}$ equals $D (t_{2}) - D (t_{1})$ . It equals the height of $D$ at $t_{2}$ only when the chosen start time is $t_{1}$ . A large positive area accumulated before $t_{1}$ can place the graph high above zero while leaving the later interval displacement unchanged. Definite-integral limits belong to the physical statement and keep a graph offset from being mistaken for an interval result.

The cumulative signed area beneath velocity becomes the position change from the stated start time. Positive velocity makes the cumulative curve rise; negative velocity makes it fall. The same zero of velocity appears as a horizontal tangent on the accumulated-displacement graph.

Cross-checking measured graph data.

Experimental graph sets should be checked in both directions. A slope estimated from the position trace predicts local velocity. A signed area from the velocity trace predicts position change. A slope from the velocity trace predicts acceleration, and an area from the acceleration trace predicts velocity change. Exact agreement is not expected when curves are drawn through independent noisy measurements, but disagreement larger than the stated reading uncertainty points to a timing offset, a calibration error, or an inappropriate smoothing method.

The comparison must be made on matching intervals. A centered velocity difference derived from position values at $t - Δ t$ and $t + Δ t$ belongs at time $t$ . An acceleration computed from two successive velocity samples belongs naturally at the midpoint between them. Shifting these derived values to an endpoint can create an apparent lag between the velocity and acceleration graphs even when the physical measurements are consistent.

Unit cancellation provides an initial audit. Position differences divided by time give velocity; velocity differences divided by time give acceleration; velocity times time gives position change. An incorrect time unit can still yield a plausible-looking number, so display the unit cancellation in a written calculation. A millisecond clock treated as seconds introduces factors of one thousand into a slope or area.

Three measured positions admit many smooth acceleration histories. Additional readings and residuals against a fitted model establish whether a constant-acceleration approximation is justified over a longer interval.

A graph-reading protocol.

An unfamiliar motion graph should first be identified by its plotted quantity and units. Mark the time interval before extracting a slope or area. Identify whether a zero is a zero of position, velocity, or acceleration; each has a different physical meaning. Use the graph relation appropriate to the vertical quantity, preserve signs throughout the computation, and state whether the result is instantaneous, average, or an estimated integral. These few steps prevent a visually neat calculation from answering the wrong kinematic question.

Time shifts and finite differences

The clock origin is a convention. Replacing $t$ with $T = t - t_{0}$ shifts every event left or right on a graph but does not change any measured velocity or acceleration. The slope of a position graph is unchanged by a horizontal translation, and the area under velocity between two physical events is unchanged when the same events are expressed with a new time coordinate. Confusion arises only when numerical time labels from one origin are mixed with intervals measured from another.

Position origins behave differently. Replacing $x$ with $X = x - x_{0}$ shifts the position graph vertically but leaves velocity, acceleration, displacement, and distance unchanged. A vertical offset can make every displayed position positive without changing whether an object reverses direction. Direction is determined by the slope of the position trace or the sign of velocity, never by whether the line lies above the plotted horizontal axis.

Velocity offsets need physical interpretation. A constant vertical shift in a velocity graph can arise when the reference frame moves at constant speed. It changes the displacement reported by that frame and may change whether the velocity crosses zero, while acceleration is unchanged. A vertical shift caused by an uncorrected sensor zero is instead an error: integrating it creates a position error that grows with time. Graph transformations should therefore be labelled as coordinate changes or calibration corrections, not treated as interchangeable drawing operations.

Finite-difference slopes and sampling limits.

Recorded positions arrive at discrete clock times, whereas the derivative $v (t) = d x / d t$ refers to an instant. A difference quotient bridges those two descriptions. For equally spaced position samples separated by $Δ t$ , the forward quotient

v_{i + 1/2} \approx \frac{x _{i + 1} - x _{i}}{Δ t}

is the average velocity across the interval. Its natural time location is the midpoint $t_{i + 1/2}$ , not the left-hand sample time. A backward quotient has the same meaning on the preceding interval. Assigning either one to an endpoint hides the half-sample timing shift and can make a derived velocity record appear to lag or lead a position record.

The centered quotient for a smooth position function

v_{i} \approx \frac{x _{i + 1} - x _{i - 1}}{2Δ t}

uses readings equally far before and after $t_{i}$ . Its estimate belongs at $t_{i}$ . The symmetric geometry also removes the first-order curvature error that affects a one-sided quotient. That improvement assumes comparable spacing and a smoothly varying path; it does not turn sparse or noisy positions into an exact velocity measurement. The position samples still supply only the secant information shown in the figure, while a tangent remains a local limiting slope.

The second derivative is more demanding. At equal time spacing, a central position estimate for acceleration is

a_{i} \approx \frac{x _{i + 1} - 2 x _{i} + x _{i - 1}}{( Δ t ) ^{2}} .

It measures how successive position increments differ. Equal increments give zero in the numerator and therefore zero estimated acceleration. Increments that grow by the same amount from one interval to the next give a constant acceleration. The estimate is assigned to the middle time $t_{i}$ because it compares the intervals on either side of that time. A calculation that reports it at $t_{i + 1}$ shifts the entire acceleration sequence by one sample.

Measurement uncertainty changes the choice of time interval. Suppose independent position readings each have uncertainty $σ_{x}$ . The standard uncertainty of a two-point velocity difference is approximately

σ_{v} \approx \frac{2 σ _{x}}{Δ t},

and the corresponding three-point acceleration difference has uncertainty of order $6 σ_{x} / (Δ t)^{2}$ . Halving the sample interval therefore doubles the noise scale in a velocity difference and roughly quadruples it in a position- derived acceleration. Faster sampling captures short changes in motion, yet a very short interval can produce a derivative dominated by digitization, marker jitter, or camera-pixel error. The interval must be chosen against both the motion time scale and the instrument resolution.

Smoothing may lower random scatter before differentiation, but it also changes the record. A moving average blurs sharp starts, stops, and collision features across several sample times. Polynomial fits and spline fits supply differentiable curves, but their derivative depends on the fitting window and model order. A graph or report should identify whether a velocity trace came from raw differences, a filtered record, or a fitted trajectory. That distinction matters most near an event whose duration is comparable with the smoothing window.

Consider position readings separated by $0.20 s$ : $x_{i - 1} = 1.06 m$ , $x_{i} = 1.42 m$ , and $x_{i + 1} = 1.90 m$ . The centered velocity at the middle reading is

v_{i} \approx \frac{1.90 - 1.06}{0.40} = 2.10 m s^{- 1} .

The second-difference acceleration is

a_{i} \approx \frac{1.90 - 2 ( 1.42 ) + 1.06}{( 0.20 ) ^{2}} = 3.0 m s^{- 2} .

Both values belong to the time of $x_{i}$ . The calculation uses three readings but does not establish a constant-acceleration law beyond that neighborhood. A longer set of points, plotted residuals, and stated instrument uncertainty determine whether the local estimate represents a stable trend or a measurement fluctuation.

Sampling also imposes an aliasing limit. If an object reverses direction between two position readings, nearly equal endpoints can conceal a substantial intervening excursion. A short secant would then be mistaken for a small velocity even though the object travelled far during the interval. Video analysis needs a frame rate that resolves the relevant change of motion and a clock synchronized with the image samples. No differentiation procedure can recover a feature absent from the record.