# A.1 Kinematics

> IB Physics Higher Level · Theme A: Space, time and motion
> Source: https://www.owlsprep.com/study/ib-physics-hl-u1-a-1-kinematics/

This sub-topic introduces core concepts to describe linear motion, including scalar/vector quantities, the kinematic (SUVAT) equations for constant acceleration, and interpretation of motion graphs. It forms the foundation for all mechanics topics in IB Physics HL.

**Prerequisites:** Basic algebraic manipulation and graph interpretation

## Learning objectives

- Distinguish between scalar and vector quantities for motion
- Calculate displacement, velocity and acceleration for linear motion
- Apply kinematic (SUVAT) equations for constant acceleration
- Interpret displacement-time and velocity-time motion graphs

## Scalar and Vector Quantities

**Scalar Quantity** — A quantity that has only magnitude (size) and no associated direction.

*Example:* Distance, speed, mass, time, energy are all scalars.

**Vector Quantity** — A quantity that has both magnitude and direction, and follows the rules of vector addition.

*Example:* Displacement, velocity, acceleration, force are all vectors.

For one-dimensional motion, we simplify vector calculations by assigning a positive sign to motion in one direction, and a negative sign to motion in the opposite direction.

**Worked example:** A hiker walks 8 km north, then turns around and walks 3 km south. Calculate the total distance travelled and the total displacement.

1. Distance is a scalar, so add all magnitudes:
2. $$8 + 3 = 11 \text{ km}$$
3. Assign north as positive, so south is negative. Displacement is a vector, so add signed values:
4. $$\text{Displacement} = (+8) + (-3) = +5 \text{ km}$$
5. Final answer: Total distance = 11 km, total displacement = 5 km north.

> **Exam tip:** Always check if a question asks for distance or displacement — they are almost always marked as separate answers.

## Core Kinematic Quantities

**Average Velocity** — Total displacement divided by total time taken for the motion

*Notation:* v_{\text{avg}}

*Example:* $v_{\text{avg}} = \frac{\Delta s}{\Delta t}$

**Acceleration** — Rate of change of velocity, equal to the gradient of a velocity-time graph

*Notation:* a

*Example:* $a = \frac{\Delta v}{\Delta t}$ for constant acceleration

For motion with constant acceleration, average velocity can also be calculated as the average of the initial and final velocity, regardless of time taken.

**Worked example:** A cyclist accelerates from rest to 12 m s⁻¹ over 6 seconds. Calculate the average velocity and acceleration of the cyclist.

1. We know initial velocity $u = 0$, final velocity $v = 12$ m s⁻¹, time $t = 6$ s.
2. Average velocity for constant acceleration:
3. $$v_{\text{avg}} = \frac{u + v}{2} = \frac{0 + 12}{2} = 6 \text{ m s}^{-1}$$
4. Acceleration is change in velocity over time:
5. $$a = \frac{v - u}{t} = \frac{12 - 0}{6} = 2 \text{ m s}^{-2}$$

## Kinematic Equations (SUVAT)

> **SUVAT Acronym**
>
> The 5 kinematic quantities are remembered by the acronym SUVAT: S = displacement, U = initial velocity, V = final velocity, A = acceleration, T = time.

For motion with constant acceleration, the four valid kinematic equations are:

$$v = u + at$$

$$s = \frac{(u + v)t}{2}$$

$$s = ut + \frac{1}{2} a t^2$$

$$v^2 = u^2 + 2as$$

These equations only work when acceleration is constant. They are used for all one-dimensional constant acceleration problems, including free fall.

**Worked example:** A stone is dropped from rest from a 45 m tall cliff. Taking $g = 9.8$ m s⁻², calculate the time taken for the stone to hit the ground.

1. Take downward as positive. List known SUVAT values: $u = 0$, $s = 45$ m, $a = 9.8$ m s⁻², $t = ?$
2. Select the equation that contains only the known values and the unknown we need to find:
3. $$s = ut + \frac{1}{2} a t^2$$
4. Substitute values, the $ut$ term cancels out because $u=0$:
5. $$45 = 0 + \frac{1}{2}(9.8)t^2 \implies t^2 = \frac{90}{9.8} \approx 9.18$$
6. Take the positive root (time cannot be negative):
7. $$t \approx 3.03 \text{ s}$$

> **Exam tip:** Always pick the equation that directly solves for your unknown, to avoid calculating unnecessary intermediate values and save exam time.

## Interpreting Motion Graphs

Motion can be represented graphically, and key kinematic quantities are found from gradients and areas under graphs, regardless of whether acceleration is constant:

- Gradient of a displacement-time ($s$-$t$) graph = instantaneous velocity
- Gradient of a velocity-time ($v$-$t$) graph = instantaneous acceleration
- Area under a velocity-time ($v$-$t$) graph = total displacement
- Area under an acceleration-time ($a$-$t$) graph = total change in velocity

**Worked example:** A velocity-time graph for a car is a straight line from $(t=0, v=0)$ to $(t=10\text{ s}, v=20 \text{ m s}^{-1})$. What is the total displacement of the car after 10 seconds?

1. Total displacement equals the area under the velocity-time graph.
2. The graph forms a right triangle with base 10 s and height 20 m s⁻¹:
3. $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 20 = 100 \text{ m}$$
4. Total displacement = 100 m.

## Common pitfalls

- **Wrong:** Assuming average speed is always equal to the magnitude of average velocity
  - Why it fails: Average speed is total distance over time, while average velocity is total displacement over time. They only match for motion with no direction change.
  - Correct: Always distinguish the two quantities when the object changes direction during motion.
- **Wrong:** Using SUVAT equations for motion with non-constant acceleration
  - Why it fails: SUVAT equations are derived specifically for constant acceleration, so they will give incorrect results for changing acceleration.
  - Correct: Use gradient and area methods on motion graphs for non-constant acceleration problems.
- **Wrong:** Mixing up gradients of displacement-time and velocity-time graphs
  - Why it fails: Students often confuse which gradient corresponds to which kinematic quantity in exams.
  - Correct: Memorise: Gradient of $s$-$t$ = $v$, gradient of $v$-$t$ = $a$.
- **Wrong:** Using an inconsistent sign convention for direction
  - Why it fails: For example, taking upward as positive but writing acceleration due to gravity as positive leads to wrong sign results.
  - Correct: Always write down your chosen positive direction at the start of any kinematics problem, and stick to it.
- **Wrong:** Keeping negative time solutions from quadratic kinematic equations
  - Why it fails: Quadratic equations give two solutions, but negative time refers to a moment before motion started, which is unphysical.
  - Correct: Always discard negative time solutions unless the question explicitly asks for time before $t=0$.

## Cheatsheet

| Quantity/Relationship | Symbol/Rule |
| --- | --- |
| Scalar | Magnitude only, no direction |
| Vector | Magnitude + direction |
| Average velocity | $\frac{\Delta s}{\Delta t}$ |
| Average acceleration | $\frac{\Delta v}{\Delta t}$ |
| $v =$ | $u + at$ |
| $s =$ | $\frac{(u+v)t}{2}$ |
| $s =$ | $ut + \frac{1}{2} a t^2$ |
| $v^2 =$ | $u^2 + 2 a s$ |
| Gradient $s$-$t$ graph | Velocity |
| Gradient $v$-$t$ graph | Acceleration |
| Area under $v$-$t$ graph | Displacement |

## What's next

A.1 Kinematics is the foundational topic for all mechanics in IB Physics HL. All subsequent motion topics build on the core concepts of displacement, velocity, acceleration and the SUVAT framework introduced here. Next, you will extend these ideas to two-dimensional motion with projectile motion, then use kinematics to understand how forces cause acceleration in Newton's laws of motion. Mastery of this sub-topic is critical for later topics including circular motion, orbital mechanics and simple harmonic motion.

- [A.2 Forces and momentum](https://www.owlsprep.com/study/ib-physics-hl-u1-a-2-forces-and-momentum/)
- [A.3 Work, energy and power](https://www.owlsprep.com/study/ib-physics-hl-u1-a-3-work-energy-and/)
- [A.4 Rotational mechanics](https://www.owlsprep.com/study/ib-physics-hl-u1-a-4-rotational-mechanics/)

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