14.14 Calculus in Kinematics

2026 Syllabus Objectives

By the end of these notes, you will be able to:

  • Apply differentiation to find velocity from displacement, and acceleration from velocity, for a particle moving in a straight line.
  • Apply integration to find velocity from acceleration, and displacement from velocity.
  • Solve kinematics problems involving both variable and constant acceleration.

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1. What is Kinematics?

Kinematics is the study of how objects move. In this topic, we look at a particle (think of it as a tiny object, like a bead) moving along a straight line. We are interested in three things:

  • Displacement — where the particle is
  • Velocity — how fast it is moving (and in which direction)
  • Acceleration — how quickly its velocity is changing

2. Key Quantities — Scalars vs Vectors

Before going further, it is important to understand the difference between two types of quantities:

QuantityTypeMeaning
DistanceScalar (size only)How far a particle has travelled in total
SpeedScalar (size only)How fast a particle is moving
Displacement (s)Vector (size + direction)How far the particle is from a fixed starting point O, measured in a specific direction
Velocity (v)Vector (size + direction)The rate at which displacement changes
Acceleration (a)Vector (size + direction)The rate at which velocity changes

💡 Plain English: A vector has a direction as well as a size. For example, if you walk 5 m to the right, your displacement is +5 m. If you walk 5 m to the left, your displacement is −5 m. The sign (+ or −) tells you the direction.

3. The Sign Convention

Because displacement, velocity, and acceleration are vectors, we use positive and negative signs to show direction:

SignMeaning (typical convention)
Positive (+)Moving in the positive direction (e.g. to the right)
Negative (−)Moving in the opposite direction (e.g. to the left)
  • If v > 0: the particle is moving in the positive direction.
  • If v < 0: the particle is moving in the negative direction.
  • If v = 0: the particle is at rest (not moving at that moment — we call this instantaneous rest).
  • If a > 0: the particle is speeding up in the positive direction (or slowing down in the negative direction).
  • If a < 0: the particle is decelerating (slowing down) in the positive direction (or speeding up in the negative direction).

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