Introduction to Kinematics

Kinematics: The branch of mathematics and physics that describes the motion of objects, focusing on displacement, velocity, acceleration, and distance , without considering the forces causing the motion.

In the IB Mathematics AI HL course, kinematics problems are solved using calculus , specifically differentiation and integration , rather than relying on the constant-acceleration formulae used in physics. This means we work with functions of time that can model variable acceleration, making the approach far more general and powerful.

The three core quantities you will work with are:

Displacement $s (t)$ , the position of an object relative to a fixed reference point (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 one-dimensional kinematics, direction is indicated by the sign of the quantity. Positive values typically mean motion to the right (or upward), and negative values mean motion to the left (or downward), depending on the context defined in the problem.

The Fundamental Calculus Relationships

The entire framework of calculus-based kinematics rests on two key relationships:

Differentiation (moving "down" the chain):
$v (t) = \frac{d s}{d t} a (t) = \frac{d v}{d t} = \frac{d ^{2} s}{d t ^{2}}$

Integration (moving "up" the chain):
$v (t) = \int a (t) d t s (t) = \int v (t) d t$

You can think of the three quantities as a chain:
$s (t) integrate ⇌ differentiate v (t) integrate ⇌ differentiate a (t)$

Analogy

Think of it like a lift (elevator). Displacement is the floor you're on, velocity is how fast the lift is moving, and acceleration is how quickly the lift is speeding up or slowing down. Differentiation takes you "up" from position to rate of change; integration brings you back "down".

Warning

When integrating to find $v (t)$ or $s (t)$ , you will always obtain a constant of integration $C$ . You must use an initial condition (e.g., $v (0)$ or $s (0)$ ) to determine $C$ . Forgetting to find $C$ is one of the most common errors in exam questions.

Free preview

9 more sections in this topic

Introduction to Kinematics

The three core quantities you will work with are:

Displacement $s (t)$ , the position of an object relative to a fixed reference point (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

The Fundamental Calculus Relationships

The entire framework of calculus-based kinematics rests on two key relationships:

Differentiation (moving "down" the chain):
$v (t) = \frac{d s}{d t} a (t) = \frac{d v}{d t} = \frac{d ^{2} s}{d t ^{2}}$

Integration (moving "up" the chain):
$v (t) = \int a (t) d t s (t) = \int v (t) d t$

You can think of the three quantities as a chain:
$s (t) integrate ⇌ differentiate v (t) integrate ⇌ differentiate a (t)$

Analogy

Warning

Free preview

AHL 5.13—Kinematic problems

Introduction to Kinematics

The Fundamental Calculus Relationships

9 more sections in this topic

AHL 5.13—Kinematic problems

Introduction to Kinematics

The Fundamental Calculus Relationships

9 more sections in this topic