Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.13
📋 Learning Intentions
- Define displacement, velocity, and acceleration as functions of time, and state the calculus relationships connecting them.
- Find velocity and acceleration by differentiating a displacement function, and find displacement by integrating a velocity function.
- Calculate total distance travelled by integrating the absolute value of velocity over a time interval.
- Distinguish between displacement and total distance travelled, and between velocity and speed.
- Determine when a particle is at rest, moving in a positive direction, or changing direction using sign analysis of velocity.
- Use the GDC to evaluate definite integrals and solve equations arising in kinematics problems.
Real-World Introduction
A sprinter crosses the finish line and immediately decelerates. A drone climbs vertically, hovers, then descends. A ball is thrown upward and falls back to the ground. In every case, the position of the object changes over time — and the rate at which that position changes is velocity, while the rate at which velocity changes is acceleration. These three quantities are linked by calculus.
Engineers designing roller coasters, physicists modelling planetary orbits, and sports scientists analysing athletic performance all use the same mathematical framework: differentiation to move from position to velocity to acceleration, and integration to move in the reverse direction. The IB kinematics topic gives you the tools to work in both directions along this chain.
🚗 Analogy — The Speedometer and Odometer A car's speedometer reads instantaneous velocity (rate of change of position). The odometer records total distance travelled — it never decreases, even when the car reverses. These two instruments measure different things, and the distinction between them is exactly the distinction between displacement and total distance in kinematics.
Core Concepts & Definitions
Motion along a straight line is described by three related functions of time: displacement \(s(t)\), velocity \(v(t)\), and acceleration \(a(t)\). Each can be obtained from the previous by differentiation, or from the next by integration. Understanding the direction of this chain — and when to use each operation — is the foundation of all kinematics problems.
🔑 Key Definition — Displacement \(s(t)\) Displacement is the signed distance of a particle from a fixed reference point (the origin) at time \(t\). It is measured in metres (m) and can be positive, negative, or zero. A positive value means the particle is to the right of (or above) the origin; a negative value means it is to the left of (or below) the