The idea

Kinematics with calculus (a → v → s)

Why integrating acceleration gives velocity and integrating velocity gives displacement, where the suvat formulas actually come from, and why they silently break the moment acceleration varies.

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Mechanics · Kinematics

Kinematics with calculus (a → v → s)

Why integrating acceleration gives velocity and integrating velocity gives displacement, where the suvat formulas actually come from, and why they silently break the moment acceleration varies.

Why it works

Up and down the chain

Velocity is defined as the rate of change of displacement, and acceleration as the rate of change of velocity:

v = \frac{ds}{dt}, \qquad a = \frac{dv}{dt}

Differentiation takes you down that chain (

s \to v \to a

: "how fast is this changing?"). So going back up the chain — from how the motion is changing to the motion itself — is integration:

v = \int a\,dt, \qquad s = \int v\,dt

The constant is physics

Each integration introduces a

+C

, and in mechanics the constant isn't abstract — it's physics. Integrating acceleration tells you how velocity changes, but not what it started at; the constant is the initial velocity. Same again for displacement: the constant is where the particle began. That's why exam questions always hand you facts like "initially at rest" or "starts at the origin" — they are the values of your constants of integration.

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