Acceleration

This lesson covers acceleration, velocity-time graphs, and the kinematic equation v² = u² + 2as — as required by the Edexcel GCSE Physics specification (1PH0), Topic 1: Key Concepts of Physics. You need to be able to calculate acceleration, interpret velocity-time graphs, and apply the kinematic equation (Higher tier).

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What Is Acceleration?

Acceleration is the rate of change of velocity. It tells you how quickly an object's velocity is changing.

acceleration = change in velocity ÷ time taken

$a = \frac{\Delta v}{t} = \frac{v - u}{t}$a=tΔv​=tv−u​

Where:

• a = acceleration (m/s²)
• v = final velocity (m/s)
• u = initial velocity (m/s)
• t = time taken (s)
• Δv = change in velocity = v − u

Key Points

• Acceleration is a vector quantity — it has both magnitude and direction.
• The SI unit of acceleration is metres per second squared (m/s²).
• A positive acceleration means the object is speeding up (in the direction of travel).
• A negative acceleration (deceleration) means the object is slowing down.

Exam Tip: Deceleration is not a separate concept — it is simply acceleration in the opposite direction to motion. In calculations, if an object slows down, the acceleration will come out as a negative number. Do not discard the negative sign.

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Uniform and Non-Uniform Acceleration

• Uniform (constant) acceleration — the velocity changes by equal amounts in equal time intervals. Example: a freely falling object near Earth's surface (ignoring air resistance) accelerates at approximately 9.8 m/s².
• Non-uniform acceleration — the velocity changes by different amounts in equal time intervals. Example: a car accelerating from traffic lights — the acceleration is greatest at first and decreases as speed increases.

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Calculating Acceleration — Worked Examples

Worked Example 1

A car accelerates from rest to 25 m/s in 10 s. Calculate its acceleration.

a = (v − u) ÷ t = (25 − 0) ÷ 10 = 2.5 m/s²

Worked Example 2

A cyclist slows down from 12 m/s to 4 m/s in 8 s. Calculate the acceleration.

a = (v − u) ÷ t = (4 − 12) ÷ 8 = −8 ÷ 8 = −1.0 m/s²

The negative sign tells us this is a deceleration.

Worked Example 3

A train has an acceleration of 0.5 m/s². If it starts at 10 m/s, what is its velocity after 20 s?

Rearrange: v = u + at = 10 + (0.5 × 20) = 10 + 10 = 20 m/s

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Velocity-Time Graphs

A velocity-time graph shows how the velocity of an object changes with time. These graphs are extremely important in GCSE Physics.

How to Read a Velocity-Time Graph

Feature of the Graph	What It Means
Horizontal line	Constant velocity (zero acceleration)
Upward-sloping straight line	Constant (uniform) acceleration
Downward-sloping straight line	Constant (uniform) deceleration
Steeper gradient	Greater acceleration
Curved line	Non-uniform (changing) acceleration
Line at v = 0	Object is stationary

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Gradient = Acceleration

The gradient of a velocity-time graph gives the acceleration:

acceleration = gradient = change in velocity ÷ change in time

$a = \frac{\Delta v}{\Delta t}$a=ΔtΔv​

Worked Example

From a velocity-time graph, the velocity increases from 5 m/s to 20 m/s in 6 seconds. What is the acceleration?

a = (20 − 5) ÷ 6 = 15 ÷ 6 = 2.5 m/s²

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Area Under the Graph = Distance

The area under a velocity-time graph gives the distance travelled (or displacement).

Calculating the Area

For straight-line sections:

• Rectangle: area = base × height = time × velocity
• Triangle: area = ½ × base × height = ½ × time × velocity

Worked Example

An object accelerates uniformly from 0 to 20 m/s in 10 s, then travels at 20 m/s for 5 s. Find the total distance.

Section 1 (triangle): area = ½ × 10 × 20 = 100 m Section 2 (rectangle): area = 5 × 20 = 100 m Total distance = 100 + 100 = 200 m

graph LR
    A["0 s: Start from rest<br/>(v = 0)"] -->|"Uniform acceleration<br/>Triangle area = 100 m"| B["10 s: Reaches 20 m/s"]
    B -->|"Constant velocity<br/>Rectangle area = 100 m"| C["15 s: Still at 20 m/s"]

    style A fill:#2c3e50,color:#fff
    style B fill:#2980b9,color:#fff
    style C fill:#27ae60,color:#fff

Exam Tip: Always check which area shape you need. A triangle has area = ½ base × height. A trapezium has area = ½(a + b) × h. Count grid squares if the shape is irregular. Show your working clearly.

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The Kinematic Equation (Higher Tier)

For objects undergoing uniform acceleration, the following equation links velocity, acceleration, and distance:

v² = u² + 2as

Where:

• v = final velocity (m/s)
• u = initial velocity (m/s)
• a = acceleration (m/s²)
• s = distance (m)

This equation is useful when you are not given the time.

Worked Example 1

A car accelerates from 10 m/s to 30 m/s with a uniform acceleration of 4 m/s². Calculate the distance over which it accelerates.

v² = u² + 2as 30² = 10² + 2 × 4 × s 900 = 100 + 8s 800 = 8s s = 100 m

Worked Example 2

A ball is dropped from rest and falls 20 m. Assuming g = 9.8 m/s² and no air resistance, calculate its final velocity.

v² = u² + 2as v² = 0² + 2 × 9.8 × 20 v² = 392 v = √392 = 19.8 m/s

Exam Tip: The equation v² = u² + 2as is on the Higher tier only. If you are entered for Higher, make sure you practise rearranging this equation for each variable. Always write down the equation, substitute, then rearrange — show your working clearly for full marks.

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Comparing Distance-Time and Velocity-Time Graphs

Feature	Distance-Time Graph	Velocity-Time Graph
Gradient gives	Speed	Acceleration
Area under graph gives	(Not applicable)	Distance travelled
Horizontal line means	Stationary	Constant velocity
Straight diagonal means	Constant speed	Constant acceleration
Curve means	Changing speed	Changing acceleration

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Summary