Position, Velocity, Acceleration

When you describe how an object moves, three quantities tell the complete story: position, velocity, and acceleration. Position s(t) tells you where the object is at time t. Velocity v(t) tells you how fast and in which direction it is moving. Acceleration a(t) tells you how quickly that velocity is changing. Together, these three quantities form a hierarchy connected by the single operation of differentiation — one of the most powerful ideas in calculus. The connection is precise and beautiful: velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time (equivalently, the second derivative of position). This means every concept you have learned about derivatives — slopes, rates of change, increasing and decreasing behavior, maxima and minima — applies directly to the physics of moving objects. Calculus was, in large part, invented to describe exactly this kind of motion. Understanding motion through derivatives allows you to answer concrete questions: When is a falling ball at its highest point? How fast is a car traveling when it hits the brakes? When is an object speeding up or slowing down? These questions are answered not by memorizing formulas but by reasoning carefully with derivatives, and mastering this connection prepares you for physics, engineering, and further mathematics.

Key takeaways
  • Position Function s(t)
  • Velocity as the First Derivative
  • Acceleration as the Second Derivative
  • Reading the Signs Together
  • Integrating Back Up the Chain

Position Function s(t)

The position function s(t) gives the location of an object along a number line (or in space) at time t. The units are typically meters (m) or feet (ft). Positive and negative values indicate direction: for example, s = +5 m might mean 5 metres to the right of the origin, while s = −3 m means 3 metres to the left. The function itself can be a polynomial, trigonometric, exponential, or any differentiable function of time.

Velocity as the First Derivative

Velocity is the instantaneous rate of change of position: v(t) = s'(t) = ds/dt. A positive velocity means the object moves in the positive direction; negative velocity means it moves in the negative direction. Speed is the absolute value |v(t)| and is never negative. The object is momentarily at rest when v(t) = 0. Finding when v(t) = 0 is often the key step in locating maximum or minimum positions.

Acceleration as the Second Derivative

Acceleration is the instantaneous rate of change of velocity: a(t) = v'(t) = s''(t) = d²s/dt². Positive acceleration means velocity is increasing; negative acceleration (deceleration) means velocity is decreasing. Critically, 'speeding up' and 'slowing down' depend on both v(t) and a(t) together: the object speeds up when v(t) and a(t) have the same sign, and slows down when they have opposite signs.

Reading the Signs Together

To fully analyze motion, construct a sign chart for both v(t) and a(t). If v(t) > 0 and a(t) > 0, the object moves in the positive direction and speeds up. If v(t) > 0 and a(t) < 0, it moves in the positive direction but slows down. If v(t) < 0 and a(t) < 0, it moves in the negative direction and speeds up. If v(t) < 0 and a(t) > 0, it moves in the negative direction but slows down. These four cases cover all possibilities on a straight-line path.

Integrating Back Up the Chain

You can also reverse the process using antiderivatives (integrals). If you know acceleration, integrate to get velocity: v(t) = ∫a(t) dt + C₁. Integrate again to get position: s(t) = ∫v(t) dt + C₂. The constants C₁ and C₂ are determined by initial conditions such as initial velocity v(0) and initial position s(0). This reverse process is essential in kinematics problems where you start from a known force or acceleration.

The Key Equation

v(t)=s′(t)=dsdt,a(t)=v′(t)=s′′(t)=d2sdt2v(t) = s'(t) = \frac{ds}{dt}, \qquad a(t) = v'(t) = s''(t) = \frac{d^2s}{dt^2}
  • s(t) — Position (metres (m) or feet (ft)): Location of the object on a number line at time t
  • v(t) — Velocity (m/s or ft/s): Rate of change of position; carries a sign indicating direction
  • a(t) — Acceleration (m/s² or ft/s²): Rate of change of velocity; positive means velocity is increasing
  • t — Time (seconds (s)): The independent variable representing elapsed time

Worked Example

Problem. A particle moves along a horizontal line with position function s(t) = t³ − 6t² + 9t − 2 (metres, t in seconds, t ≥ 0). (a) Find the velocity and acceleration functions. (b) When is the particle at rest? (c) When is the particle speeding up?

v(t)=3t2−12t+9,  a(t)=6t−12,  at rest: t=1,3 s,  speeding up: t∈(1,2)∪(3,∞)v(t)=3t^2-12t+9,\; a(t)=6t-12,\; \text{at rest: } t=1,3\text{ s},\; \text{speeding up: } t\in(1,2)\cup(3,\infty)

Answer. Velocity v(t) = 3t²−12t+9; Acceleration a(t) = 6t−12; at rest at t = 1 s and t = 3 s; speeding up on (1, 2) and (3, ∞).

Figure
Diagram
Position, Velocity, and Acceleration (t³−6t²+9t−2)ty012345v=0v=0s(t) positionv(t) velocitya(t) accel.0
Origins · 1 minIsaac Newton and how this was discovered
IN
Isaac Newton
1643–1727 · English

Newton invented calculus (which he called 'the method of fluxions') precisely to describe motion. He defined velocity as the 'fluxion' of position and used second derivatives to formulate his Second Law of Motion, F = ma, linking acceleration directly to force. His 1687 masterwork Principia Mathematica unified terrestrial and celestial mechanics using these derivative-based concepts. Newton reportedly kept his calculus discoveries secret for years, causing a bitter priority dispute with the German mathematician Gottfried Wilhelm Leibniz, who independently developed calculus around the same time and whose notation (dy/dx, ∫) is the one we still use today.

  • The Derivative: Definition and Notation — Velocity and acceleration are defined as derivatives; a solid understanding of derivative notation and limits is essential before applying them to motion.
  • The Chain Rule — Many realistic position functions involve compositions (e.g., sin(3t), e^(−kt)) requiring the chain rule to compute velocity and acceleration correctly.
  • Antiderivatives and Indefinite Integrals — Recovering position from velocity or velocity from acceleration requires integration, the reverse operation of differentiation.
  • Increasing, Decreasing, and the First Derivative Test — Determining when an object moves in the positive or negative direction is equivalent to finding intervals where the position function is increasing or decreasing.
  • Concavity and the Second Derivative Test — Acceleration is the second derivative of position, so the concavity of the position graph directly describes whether the object is speeding up or slowing down.