How Are Displacement, Velocity, and Acceleration Related Through Calculus?

Velocity is the rate of change of displacement, while acceleration is the rate of change of velocity. Therefore, differentiation moves from displacement to velocity to acceleration, while integration reverses this process.

This is AHL 5.13: Kinematic problems, so it is HL-only content. It may appear in any HL paper style, and a GDC is available throughout.

The Calculus Relationships

Let displacement from a fixed origin at time $t$ be $s(t)$. Then

$$v(t)=\frac{ds}{dt}=s'(t)$$

and

$$a(t)=\frac{dv}{dt}=\frac{d^2s}{dt^2}=s''(t).$$

Integration reverses these relationships:

$$v(t)=\int a(t)\,dt$$

and

$$s(t)=\int v(t)\,dt.$$

An initial condition is needed to determine the constant of integration.

For example, suppose

$$s(t)=t^3-6t^2+9t.$$

Differentiating gives

$$v(t)=3t^2-12t+9,$$

and differentiating again gives

$$a(t)=6t-12.$$

At $t=2$, $v(2)=-3$, so the particle is moving in the negative direction, while $a(2)=0$.

Graphically, the gradient of a displacement-time graph gives velocity, and the gradient of a velocity-time graph gives acceleration. A horizontal section on a displacement-time graph means the particle is instantaneously at rest. On a velocity-time graph, signed area represents displacement; area below the time axis contributes negatively.

Conversely, if $a(t)=6t-12$, then integrating gives $v(t)=3t^2-12t+C$. If , then , recovering . Integrating once more gives ; the condition gives . These constants encode the particle's initial velocity and initial displacement. Without initial conditions, calculus determines only a family of possible velocity or displacement functions, not one unique motion.

A common misconception is that distance and displacement are interchangeable. Over $t_1\leq t\leq t_2$, displacement is $\int_{t_1}^{t_2}v(t)\,dt$, but total distance is ; split the integral wherever .

Exam Technique

Write the derivative or integral relationship before using the GDC. Check whether the question asks for displacement, distance, velocity, or speed, where speed is (|v(t)|), and include appropriate units in the final answer.