Kinematics

IB Physics SL· 1.1 Measurements and uncertainties 1.2 Kinematics· 15 min read

1. Core Kinematic Quantities: Scalars vs Vectors★☆☆☆☆⏱ 10 min

Kinematics is the branch of mechanics that describes the motion of objects without reference to the forces that cause motion. All kinematic quantities fall into one of two categories: scalar or vector.

📘 Definition

Scalar and Vector Quantities

A scalar quantity has only magnitude (a numerical value and unit) but no direction. A vector quantity has both magnitude and direction, which is required to fully describe it.

Example:

5 m/s is a scalar speed; 5 m/s north is a vector velocity

Common scalars: distance, speed, mass, energy, time
Common vectors: displacement, velocity, acceleration, force, momentum

📐 Worked Example

A runner completes one full lap of a 400 m circular running track. What is (a) the total distance travelled, and (b) the magnitude of the runner's total displacement?

1
Recall the definitions of distance and displacement. Distance is the total length of the path travelled, while displacement is the change in position from the starting point to the end point.
2
For part (a): One full lap covers the entire 400 m track, so total distance travelled is:
3
$d = 400 \text{ m}$
4
For part (b): After one full lap, the runner finishes at the exact point they started from, so final position equals initial position:
5
$\Delta s = s_{\text{final}} - s_{\text{initial}} = 0 \text{ m}$
6
The magnitude of displacement is therefore 0 m.

Exam tip:

Exam questions frequently switch between distance/displacement and speed/velocity to test if you can distinguish between the two quantities. Always read the question carefully.

2. Average vs Instantaneous Quantities★★☆☆☆⏱ 15 min

A common source of confusion in exams is the difference between average and instantaneous kinematic values. Average quantities are calculated over a finite time interval, while instantaneous quantities are measured at a single moment in time.

📘 Definition

Average Velocity

\bar{v} = \frac{\Delta s}{\Delta t}

Average velocity for a time interval is equal to the total displacement divided by the total time taken for the interval. Average speed is total distance divided by total time.

Example:

Travelling 120 m east in 10 s gives an average velocity of 12 m/s east.

📐 Worked Example

A car drives 100 km north from town A to town B in 2 hours, then immediately drives back to town A, taking another 2 hours. What is the average velocity of the car for the entire trip?

1
Start by identifying the total displacement for the entire journey. The car starts at A and ends at A, so:
2
$\Delta s = 0 \text{ km}$
3
Total time for the journey is \Delta t = 2 + 2 = 4 \text{ h}.
4
Use the formula for average velocity:
5
$\bar{v} = \frac{\Delta s}{\Delta t} = \frac{0}{4} = 0 \text{ km/h}$
6
Note: average speed for the trip is \frac{200 \text{ km}}{4 \text{ h}} = 50 \text{ km/h}, which is not equal to average velocity.

3. Kinematic Equations for Constant Acceleration★★☆☆☆⏱ 20 min

✓ Calculator OK

For motion with uniform (constant) acceleration in a straight line, we can derive four core equations that relate the five kinematic quantities: initial velocity ( $u$ ), final velocity ( $v$ ), acceleration ( $a$ ), displacement ( $s$ ), and time ( $t$ ).

🔬 Derivation

Goal:

Derive the first kinematic equation $v = u + a t$ from the definition of acceleration

Starting from:

Acceleration is defined as rate of change of velocity, and is constant for uniform motion

1
Write the definition of constant acceleration:
2
$a = \frac{v - u}{t}$
3
Multiply both sides by $t$ :
4
$at = v - u$
5
Rearrange to isolate $v$ :

Result:

v = u + at

📐 Worked Example

A ball is dropped from rest from a height of 20 m above the ground. Acceleration due to gravity is $g = 9.8 m/s^{2}$ . Calculate the speed of the ball just before it hits the ground, and the time taken to fall.

1
Assign downward as the positive direction, then list known quantities:
2
Initial velocity $u = 0$ (dropped from rest), displacement $s = 20 m$ , acceleration $a = 9.8 m/s^{2}$ .
3
To find final velocity $v$ , use the equation that does not require time $t$ :
4
$v^2 = u^2 + 2as$
5
Substitute values:
6
$v^2 = 0^2 + 2(9.8)(20) = 392$
7
Take the positive root (since velocity is downward, matching our positive direction):
8
$v = \sqrt{392} \approx 19.8 \text{ m/s}$
9
To find time $t$ , rearrange $v = u + a t$ :
10
$t = \frac{v - u}{a} = \frac{19.8 - 0}{9.8} \approx 2.02 \text{ s}$

Exam tip:

Always assign a positive direction before starting any calculation, and check that all quantities have the correct sign matching this direction.

4. Common Pitfalls

Wrong move:

Using total distance instead of displacement to calculate average velocity

Why:

Average velocity depends only on net change in position, not the total length of the path travelled

Correct move:

Always use displacement (final position minus initial position) in the average velocity formula

Wrong move:

Forgetting to assign the correct sign to acceleration for upward motion

Why:

If you take upward as positive, acceleration due to gravity is negative, not positive

Correct move:

Assign a positive direction first, then add correct signs to all quantities before substituting into equations

Wrong move:

Using constant acceleration kinematic equations for non-constant acceleration

Why:

The four kinematic equations are only valid when acceleration does not change

Correct move:

For changing acceleration, use gradient/area of velocity-time graphs instead of the kinematic equations

Wrong move:

Keeping both positive and negative roots from $v^{2} = u^{2} + 2 a s$

Why:

Most problems only have one physically meaningful solution, the other will not match the context

Correct move:

Discard the root that does not fit the problem context (e.g. negative speed, negative time)

5. Quick Reference Cheatsheet

Equation	Used when...
v = u + at	Displacement $s$ not required/given
s = \frac{1}{2}(u+v)t	Acceleration $a$ not required/given
s = ut + \frac{1}{2}at^2	Final velocity $v$ not required/given
v^2 = u^2 + 2as	Time $t$ not required/given

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Scalar vs vector distinction
2024 · 2
Free fall kinematics calculation
2023 · 1
Average velocity calculation

What's Next

Kinematics is the foundation for all mechanics topics in IB Physics SL. Mastery of the core quantities and equations here is required for every subsequent mechanics topic. Next, you will learn how to analyze motion using displacement-time and velocity-time graphs, which are used to solve problems with non-constant acceleration that kinematic equations cannot handle. This topic also directly underpins projectile motion, where you split motion into independent horizontal and vertical components, and later dynamics, where you connect acceleration to the forces that cause motion.