3.2.1 Language of kinematics

What you'll learn

How to distinguish position, displacement, distance and distance travelled.
How velocity differs from speed.
What acceleration means, including how its sign should be interpreted.
How an equation of motion describes an object's movement over time.

Describing motion

Kinematics is the study of motion without considering the forces that cause it.

A moving object is often modelled as a particle. This means that its size and shape are ignored, so its position can be represented by a single point. This is a mathematical simplification: the real object still has dimensions.

Before describing motion along a straight line, you must choose:

an origin, from which positions are measured;
a positive direction, such as rightwards or upwards;
a unit of length, usually metres;
a time variable, usually $t$ , measured in seconds.

Key Idea

Choose a positive direction

Directions are represented using signs. Motion in the chosen positive direction has a positive value, while motion in the opposite direction has a negative value.

Position

Definition

Position

The position of a particle is its location relative to a chosen origin. For motion along a straight line, position is often represented by a coordinate such as $x$ .

For example, if rightwards is positive, a particle at $x = 5$ is 5 m to the right of the origin. A particle at $x = - 3$ is 3 m to the left of the origin.

Position can therefore be positive, negative or zero. A negative position does not mean that the particle has moved backwards; it only tells you which side of the origin the particle occupies.

Common Mistake

Confusing position with direction of motion

The sign of a particle's position does not tell you which way it is moving. A particle at $x = - 4$ could be moving either left or right.

Displacement and distance

These terms sound similar, but they describe different quantities.

Definition

Displacement

Displacement is the change in position of a particle. It has both magnitude and direction, so it is a vector quantity.

If the initial position is $x_1$ and the final position is $x_2$ , then

displacement=x2−x1.\text{displacement}=x_2-x_1.

The sign of the answer gives the direction.

Definition

Distance

The distance between two positions is the magnitude of their separation. It is a scalar quantity, so it has magnitude but no direction and cannot be negative.

For positions $x_1$ and $x_2$ ,

positions=∣x2−x1∣.\text{distance between the positions}=\left|x_2-x_1\right|.

Here, the vertical bars mean absolute value, which makes the result non-negative.

Definition

Distance travelled

The distance travelled is the total length of the route followed by the particle. Every part of the journey contributes positively, regardless of direction.

The diagram shows why displacement and distance travelled can be different.

Motion from 2 m to 8 m and back to 5 m, showing a displacement of positive 3 m and a distance travelled of 9 m

Example

Comparing displacement and distance travelled

A particle begins at $x = 2$ m, moves to $x = 8$ m, and then moves back to $x = 5$ m.

Calculate the displacement using only the initial and final positions:
$m.\text{displacement}=5-2=3\text{ m}.$
The displacement is therefore 3 m in the positive direction.
Split the route into its two parts. The first part has length $8 - 2 = 6$ m, while the return part has length $8 - 5 = 3$ m.
Add these lengths to obtain the distance travelled:
$m.\text{distance travelled}=6+3=9\text{ m}.$

Key Idea

Route versus overall change

Displacement depends only on the initial and final positions. Distance travelled depends on the entire route.

Velocity and speed

Definition

Velocity

Velocity is the rate of change of displacement with respect to time. It is a vector quantity, so its sign indicates direction.

The average velocity over a time interval is

taken.\text{average velocity}=\frac{\text{displacement}}{\text{time taken}}.

Velocity is normally measured in m s⁻¹.

Definition

Speed

Speed is the rate at which distance is travelled. It is a scalar quantity and is never negative.

The average speed over a complete journey is

taken.\text{average speed}=\frac{\text{total distance travelled}}{\text{total time taken}}.

At a particular instant, speed is the magnitude of velocity:

speed=∣velocity∣.\text{speed}=\left|\text{velocity}\right|.

Example

Calculating average velocity and average speed

A runner travels 60 m east and then 20 m west in a total time of 16 s. Take east as positive.

The overall displacement is
$m.60-20=40\text{ m}.$
Therefore,
$s−1\text{average velocity}=\frac{40}{16}=2.5\text{ m s}^{-1}$
eastwards.
The total distance travelled is
$m.60+20=80\text{ m}.$
Hence,
$s−1.\text{average speed}=\frac{80}{16}=5\text{ m s}^{-1}.$

Common Mistake

Using distance in average velocity

Average velocity uses displacement, while average speed uses total distance travelled. They are equal only when there is no reversal of direction.

Acceleration

Definition

Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity and is normally measured in m s⁻².

The average acceleration over a time interval is

taken=v−ut,\text{average acceleration} =\frac{\text{change in velocity}}{\text{time taken}} =\frac{v-u}{t},

where $u$ is the initial velocity and $v$ is the final velocity.

Because velocity includes direction, acceleration can occur when:

the particle speeds up;
the particle slows down;
the particle changes direction.

Example

Finding acceleration from signed velocities

A particle's velocity changes from 7 m s⁻¹ to $- 5$ m s⁻¹ in 3 s.

Keep the negative sign because it shows that the final velocity is in the opposite direction.
Find the change in velocity:
$s−1.v-u=-5-7=-12\text{ m s}^{-1}.$
Divide by the time taken:
$s−2.\text{average acceleration} =\frac{-12}{3} =-4\text{ m s}^{-2}.$
The acceleration is 4 m s⁻² in the negative direction.

Does negative acceleration mean slowing down?

Not necessarily. To decide whether a particle is speeding up or slowing down, compare the signs of its velocity and acceleration:

Same sign: the magnitude of the velocity increases, so the particle speeds up.
Opposite signs: the magnitude of the velocity decreases, so the particle slows down.

For example, if both velocity and acceleration are negative, the particle is moving in the negative direction and speeding up.

Common Mistake

Negative acceleration is not always deceleration

A negative acceleration means acceleration in the chosen negative direction. It means slowing down only when the velocity is positive.

Equations of motion

Definition

Equation of motion

An equation of motion is an equation giving a particle's position or displacement in terms of time.

For example,

x=2t^2-3t+4

is an equation of motion. It gives the position $x$ in metres at time $t$ seconds. Substituting a value of $t$ tells you where the particle is at that time.

If position is given as a differentiable function $x (t)$ , then instantaneous velocity and acceleration are found using differentiation:

v=dxdt,a=dvdt=d2xdt2.v=\frac{\mathrm{d}x}{\mathrm{d}t}, \qquad a=\frac{\mathrm{d}v}{\mathrm{d}t} =\frac{\mathrm{d}^2x}{\mathrm{d}t^2}.

Example

Interpreting an equation of motion

A particle moves along a straight line with position

x=t^2-6t+5,

where $x$ is measured in metres and $t$ in seconds.

At $t = 0$ , the particle's position is
$m.x=0^2-6(0)+5=5\text{ m}.$
Differentiate the equation of motion to find the velocity:
$v=dxdt=2t−6.v=\frac{\mathrm{d}x}{\mathrm{d}t}=2t-6.$
At $t = 4$ , the velocity is
$s−1.v=2(4)-6=2\text{ m s}^{-1}.$
The positive sign means that the particle is moving in the chosen positive direction.
Differentiate the velocity to find the acceleration:
$s−2.a=\frac{\mathrm{d}v}{\mathrm{d}t}=2\text{ m s}^{-2}.$

Tip

Check dimensions and units

Position and displacement use metres, velocity uses m s⁻¹, and acceleration uses m s⁻². Correct units provide a quick check that you have calculated the intended quantity.

Exam technique

In the exam

State your positive direction clearly whenever directions are involved, then use signed quantities consistently.
For displacement or average velocity, use final position minus initial position; for distance travelled or average speed, account for every part of the route.
Interpret signs in context: a negative value indicates direction, not automatically a negative distance or a particle slowing down.
Include SI units and give vector answers with either a sign or a stated direction.

Self review

Check yourself

A particle moves from $x = - 3$ m to $x = 5$ m and then returns to $x = 1$ m. What are its displacement and distance travelled?
Can a particle have negative velocity but positive acceleration? If so, what happens to its speed initially?
If $x=3t^2-4t+1$ , how would you find the particle's velocity and acceleration?

What you'll learn

How to distinguish position, displacement, distance and distance travelled.
How velocity differs from speed.
What acceleration means, including how its sign should be interpreted.
How an equation of motion describes an object's movement over time.

Describing motion

Kinematics is the study of motion without considering the forces that cause it.

Before describing motion along a straight line, you must choose:

an origin, from which positions are measured;
a positive direction, such as rightwards or upwards;
a unit of length, usually metres;
a time variable, usually $t$ , measured in seconds.

Key Idea

Choose a positive direction

Directions are represented using signs. Motion in the chosen positive direction has a positive value, while motion in the opposite direction has a negative value.

Position

Definition

Position

The position of a particle is its location relative to a chosen origin. For motion along a straight line, position is often represented by a coordinate such as $x$ .

For example, if rightwards is positive, a particle at $x = 5$ is 5 m to the right of the origin. A particle at $x = - 3$ is 3 m to the left of the origin.

Position can therefore be positive, negative or zero. A negative position does not mean that the particle has moved backwards; it only tells you which side of the origin the particle occupies.

Common Mistake

Confusing position with direction of motion

The sign of a particle's position does not tell you which way it is moving. A particle at $x = - 4$ could be moving either left or right.

Displacement and distance

These terms sound similar, but they describe different quantities.

Definition

Displacement

Displacement is the change in position of a particle. It has both magnitude and direction, so it is a vector quantity.

If the initial position is $x_1$ and the final position is $x_2$ , then

displacement=x2−x1.\text{displacement}=x_2-x_1.

The sign of the answer gives the direction.

Definition

Distance

The distance between two positions is the magnitude of their separation. It is a scalar quantity, so it has magnitude but no direction and cannot be negative.

For positions $x_1$ and $x_2$ ,

positions=∣x2−x1∣.\text{distance between the positions}=\left|x_2-x_1\right|.

Here, the vertical bars mean absolute value, which makes the result non-negative.

Definition

Distance travelled

The distance travelled is the total length of the route followed by the particle. Every part of the journey contributes positively, regardless of direction.

The diagram shows why displacement and distance travelled can be different.

Motion from 2 m to 8 m and back to 5 m, showing a displacement of positive 3 m and a distance travelled of 9 m

Example

Comparing displacement and distance travelled

A particle begins at $x = 2$ m, moves to $x = 8$ m, and then moves back to $x = 5$ m.

Calculate the displacement using only the initial and final positions:
$m.\text{displacement}=5-2=3\text{ m}.$
The displacement is therefore 3 m in the positive direction.
Split the route into its two parts. The first part has length $8 - 2 = 6$ m, while the return part has length $8 - 5 = 3$ m.
Add these lengths to obtain the distance travelled:
$m.\text{distance travelled}=6+3=9\text{ m}.$

Key Idea

Route versus overall change

Displacement depends only on the initial and final positions. Distance travelled depends on the entire route.

Velocity and speed

Definition

Velocity

Velocity is the rate of change of displacement with respect to time. It is a vector quantity, so its sign indicates direction.

The average velocity over a time interval is

taken.\text{average velocity}=\frac{\text{displacement}}{\text{time taken}}.

Velocity is normally measured in m s⁻¹.

Definition

Speed

Speed is the rate at which distance is travelled. It is a scalar quantity and is never negative.

The average speed over a complete journey is

taken.\text{average speed}=\frac{\text{total distance travelled}}{\text{total time taken}}.

At a particular instant, speed is the magnitude of velocity:

speed=∣velocity∣.\text{speed}=\left|\text{velocity}\right|.

Example

Calculating average velocity and average speed

A runner travels 60 m east and then 20 m west in a total time of 16 s. Take east as positive.

The overall displacement is
$m.60-20=40\text{ m}.$
Therefore,
$s−1\text{average velocity}=\frac{40}{16}=2.5\text{ m s}^{-1}$
eastwards.
The total distance travelled is
$m.60+20=80\text{ m}.$
Hence,
$s−1.\text{average speed}=\frac{80}{16}=5\text{ m s}^{-1}.$

Common Mistake

Using distance in average velocity

Average velocity uses displacement, while average speed uses total distance travelled. They are equal only when there is no reversal of direction.

Acceleration

Definition

Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity and is normally measured in m s⁻².

The average acceleration over a time interval is

taken=v−ut,\text{average acceleration} =\frac{\text{change in velocity}}{\text{time taken}} =\frac{v-u}{t},

where $u$ is the initial velocity and $v$ is the final velocity.

Because velocity includes direction, acceleration can occur when:

the particle speeds up;
the particle slows down;
the particle changes direction.

Example

Finding acceleration from signed velocities

A particle's velocity changes from 7 m s⁻¹ to $- 5$ m s⁻¹ in 3 s.

Keep the negative sign because it shows that the final velocity is in the opposite direction.
Find the change in velocity:
$s−1.v-u=-5-7=-12\text{ m s}^{-1}.$
Divide by the time taken:
$s−2.\text{average acceleration} =\frac{-12}{3} =-4\text{ m s}^{-2}.$
The acceleration is 4 m s⁻² in the negative direction.

Does negative acceleration mean slowing down?

Not necessarily. To decide whether a particle is speeding up or slowing down, compare the signs of its velocity and acceleration:

Same sign: the magnitude of the velocity increases, so the particle speeds up.
Opposite signs: the magnitude of the velocity decreases, so the particle slows down.

For example, if both velocity and acceleration are negative, the particle is moving in the negative direction and speeding up.

Common Mistake

Negative acceleration is not always deceleration

A negative acceleration means acceleration in the chosen negative direction. It means slowing down only when the velocity is positive.

Equations of motion

Definition

Equation of motion

An equation of motion is an equation giving a particle's position or displacement in terms of time.

For example,

x=2t^2-3t+4

is an equation of motion. It gives the position $x$ in metres at time $t$ seconds. Substituting a value of $t$ tells you where the particle is at that time.

If position is given as a differentiable function $x (t)$ , then instantaneous velocity and acceleration are found using differentiation:

v=dxdt,a=dvdt=d2xdt2.v=\frac{\mathrm{d}x}{\mathrm{d}t}, \qquad a=\frac{\mathrm{d}v}{\mathrm{d}t} =\frac{\mathrm{d}^2x}{\mathrm{d}t^2}.

Example

Interpreting an equation of motion

A particle moves along a straight line with position

x=t^2-6t+5,

where $x$ is measured in metres and $t$ in seconds.

At $t = 0$ , the particle's position is
$m.x=0^2-6(0)+5=5\text{ m}.$
Differentiate the equation of motion to find the velocity:
$v=dxdt=2t−6.v=\frac{\mathrm{d}x}{\mathrm{d}t}=2t-6.$
At $t = 4$ , the velocity is
$s−1.v=2(4)-6=2\text{ m s}^{-1}.$
The positive sign means that the particle is moving in the chosen positive direction.
Differentiate the velocity to find the acceleration:
$s−2.a=\frac{\mathrm{d}v}{\mathrm{d}t}=2\text{ m s}^{-2}.$

Tip

Check dimensions and units

Position and displacement use metres, velocity uses m s⁻¹, and acceleration uses m s⁻². Correct units provide a quick check that you have calculated the intended quantity.

Exam technique

In the exam

State your positive direction clearly whenever directions are involved, then use signed quantities consistently.
For displacement or average velocity, use final position minus initial position; for distance travelled or average speed, account for every part of the route.
Interpret signs in context: a negative value indicates direction, not automatically a negative distance or a particle slowing down.
Include SI units and give vector answers with either a sign or a stated direction.

Self review

Check yourself

A particle moves from $x = - 3$ m to $x = 5$ m and then returns to $x = 1$ m. What are its displacement and distance travelled?
Can a particle have negative velocity but positive acceleration? If so, what happens to its speed initially?
If $x=3t^2-4t+1$ , how would you find the particle's velocity and acceleration?

What you'll learn

Describing motion

Position

Displacement and distance

Velocity and speed

Acceleration

Does negative acceleration mean slowing down?

Equations of motion

Review 3.2.1 Language of kinematics by teaching Genie

3.2.1 Language of kinematics

What you'll learn

Describing motion

Position

Displacement and distance

Velocity and speed

Acceleration

Does negative acceleration mean slowing down?

Equations of motion

Review 3.2.1 Language of kinematics by teaching Genie

3.2.1 Language of kinematics

What you'll learn

Describing motion

Position

Displacement and distance

Velocity and speed

Acceleration

Does negative acceleration mean slowing down?

Equations of motion

Review 3.2.1 Language of kinematics by teaching Genie

3.2.1 Language of kinematics Revision Guide

3.2.1 Language of kinematics

What you'll learn

Describing motion

Position

Displacement and distance

Velocity and speed

Acceleration

Does negative acceleration mean slowing down?

Equations of motion

Review 3.2.1 Language of kinematics by teaching Genie

3.2.1 Language of kinematics Revision Guide