Position, velocity, and acceleration as functions

Motion is not three unrelated numbers. Position tells you where an object is; velocity tells you how position changes; acceleration tells you how velocity changes.

Think like a programmer

Store the smallest state needed for the next update. For constant acceleration, time plus initial conditions lets you derive position and velocity instead of storing stale copies of both.

Model checklist

Inputs: Initial position in metres, initial velocity in m/s, acceleration in m/s², and time in seconds.
State: Time and the fixed initial conditions.
Rule: Evaluate the constant-acceleration functions.
Output: Position and velocity at the selected time.
Check: At time zero, values equal their initial conditions.

\[x(t)=x_0+v_0t+\\frac{1}{2}at^2,\\qquad v(t)=v_0+at\]

Loading the interactive visual. The lesson text and model remain available while it starts.

const position = initialPosition + initialVelocity * time + 0.5 * acceleration * time ** 2;
const velocity = initialVelocity + acceleration * time;

Try this experiment

Prediction: At time zero, the displayed position and velocity match the values in the code.

Move time to 0, then to 4 seconds. Identify which displayed quantity changes linearly and which has a squared-time term.

Where this model breaks

Constant acceleration is an idealization. Drag, changing forces, or a collision make acceleration depend on state or time and need an update loop or a new equation.

Summary

Model each quantity as a function with units. Derive values from shared state when possible, then test the initial condition directly.

Glossary

Position: location relative to a chosen origin.
Velocity: rate of change of position.
Acceleration: rate of change of velocity.

Self-check

Which value must match at time zero?
Which function contains a squared-time term?
What force could break the constant-acceleration assumption?

Sources

OpenStax, University Physics Volume 1, motion chapter.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Position, Velocity, and Acceleration as Functions, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In One-Dimensional Motion, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Position, Velocity, and Acceleration as Functions as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Position, Velocity, and Acceleration as Functions into a test

Scrub time through a moving object and connect its position, velocity, and acceleration to one shared state model.

Name the inputs and units that the one-dimensional motion model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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