What is Kinematics?

Kinematics

The branch of mathematics (and physics) that describes the motion of objects, focusing on displacement, velocity, acceleration, and distance — without reference to the forces causing the motion.

In IB Math AA SL, kinematics problems are a direct application of differentiation and integration. The motion of a particle along a straight line is described using three interconnected quantities:

Displacement $s (t)$ — the position of the particle relative to a fixed origin (a vector quantity)
Velocity $v (t)$ — the rate of change of displacement (a vector quantity)
Acceleration $a (t)$ — the rate of change of velocity (a vector quantity)

Note

In IB kinematics, motion is almost always along a straight line. The sign of $v$ and $s$ indicates direction — typically positive means right/forward, negative means left/backward.

The Fundamental Relationships

The three kinematic quantities are linked through calculus. These two relationships are the backbone of every kinematics problem:

Differentiation (going "forward"):
$v (t) = \frac{d s}{d t} a (t) = \frac{d v}{d t} = \frac{d ^{2} s}{d t ^{2}}$

Integration (going "backward"):
$s (t) = \int v (t) d t v (t) = \int a (t) d t$

You can think of it as a chain:

$s (t) differentiate v (t) differentiate a (t)$
$a (t) integrate v (t) integrate s (t)$

The Fundamental Relationships

Exam Tip

Every time you integrate, you introduce a constant of integration $C$ . Always use an initial condition — such as $s (0)$ or $v (0)$ — to find $C$ .

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What is Kinematics?

Kinematics

The branch of mathematics (and physics) that describes the motion of objects, focusing on displacement, velocity, acceleration, and distance — without reference to the forces causing the motion.

Displacement $s (t)$ — the position of the particle relative to a fixed origin (a vector quantity)
Velocity $v (t)$ — the rate of change of displacement (a vector quantity)
Acceleration $a (t)$ — the rate of change of velocity (a vector quantity)

Note

In IB kinematics, motion is almost always along a straight line. The sign of $v$ and $s$ indicates direction — typically positive means right/forward, negative means left/backward.

The Fundamental Relationships

The three kinematic quantities are linked through calculus. These two relationships are the backbone of every kinematics problem:

Differentiation (going "forward"):
$v (t) = \frac{d s}{d t} a (t) = \frac{d v}{d t} = \frac{d ^{2} s}{d t ^{2}}$

Integration (going "backward"):
$s (t) = \int v (t) d t v (t) = \int a (t) d t$

You can think of it as a chain:

$s (t) differentiate v (t) differentiate a (t)$
$a (t) integrate v (t) integrate s (t)$

The Fundamental Relationships

Exam Tip

Every time you integrate, you introduce a constant of integration $C$ . Always use an initial condition — such as $s (0)$ or $v (0)$ — to find $C$ .

Free preview

SL 5.9—Kinematics problems

What is Kinematics?

The Fundamental Relationships

9 more sections in this topic

SL 5.9—Kinematics problems

What is Kinematics?

The Fundamental Relationships

9 more sections in this topic