Velocity and Acceleration

This lesson covers the difference between velocity and speed, the acceleration equation, velocity–time graphs and the area-under-the-graph method, as required by the Edexcel GCSE Combined Science specification (1SC0). Understanding acceleration is essential for analysing motion.

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Velocity vs Speed

• Speed is a scalar — it has magnitude only. It tells you how fast an object is moving.
• Velocity is a vector — it has magnitude and direction.

An object moving at constant speed in a circle has a changing velocity because its direction is constantly changing, even though its speed stays the same.

Quantity	Type	Includes Direction?
Speed	Scalar	No
Velocity	Vector	Yes

Example

A car travelling at 20 m/s due north has:

• A speed of 20 m/s
• A velocity of 20 m/s north

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Acceleration

Acceleration is the rate of change of velocity. It is a vector quantity.

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

where:

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

A positive acceleration means the object is speeding up (in the direction of motion). A negative acceleration (deceleration) means the object is slowing down.

Exam Tip: Deceleration does not mean the acceleration is in the opposite direction in every case. Deceleration means the object is slowing down — the acceleration opposes the direction of motion. Always be careful with signs.

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Worked Examples

Example 1: A car accelerates from rest to 20 m/s in 8 s. Find the acceleration.

$a = \frac{v - u}{t} = \frac{20 - 0}{8} = 2.5 \text{ m/s}^2$a=tv−u​=820−0​=2.5 m/s2

Example 2: A cyclist slows from 15 m/s to 5 m/s in 4 s. Find the acceleration.

$a = \frac{v - u}{t} = \frac{5 - 15}{4} = \frac{-10}{4} = -2.5 \text{ m/s}^2$a=tv−u​=45−15​=4−10​=−2.5 m/s2

The negative sign indicates deceleration.

Example 3: A train has an acceleration of 0.5 m/s² and starts from rest. What is its velocity after 40 s?

Rearrange: $v = u + at = 0 + 0.5 \times 40 = 20 \text{ m/s}$v=u+at=0+0.5×40=20 m/s

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Velocity–Time Graphs

A velocity–time (v–t) graph shows how the velocity of an object changes over time.

Feature	What It Tells You
Horizontal line	Constant velocity (no acceleration)
Straight line sloping upward	Constant acceleration
Straight line sloping downward	Constant deceleration
Steeper gradient	Greater acceleration
Line at v = 0	Object is stationary

Gradient = Acceleration

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

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

Worked Example

A v–t graph shows velocity increasing from 10 m/s to 30 m/s over 5 s. What is the acceleration?

$a = \frac{30 - 10}{5} = \frac{20}{5} = 4 \text{ m/s}^2$a=530−10​=520​=4 m/s2

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Area Under a Velocity–Time Graph

The area under a velocity–time graph equals the distance travelled.

For a straight-line graph:

• Rectangle area = base × height
• Triangle area = ½ × base × height
• Trapezium area = ½ × (a + b) × h

Worked Example

An object accelerates uniformly from 0 to 30 m/s in 15 s. What distance does it cover?

The v–t graph is a triangle:

$d = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 15 \times 30 = 225 \text{ m}$d=21​×base×height=21​×15×30=225 m

Exam Tip: When a v–t graph has several sections, split it into rectangles and triangles, calculate each area separately, and then add them together for the total distance.

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Comparing Distance–Time and Velocity–Time Graphs

Feature	Distance–Time Graph	Velocity–Time Graph
Gradient gives	Speed	Acceleration
Horizontal line means	Stationary	Constant velocity
Area under gives	(not used)	Distance travelled
Straight line up means	Constant speed	Constant acceleration

graph LR
    subgraph "Distance-Time"
        A["gradient = speed"]
    end
    subgraph "Velocity-Time"
        B["gradient = acceleration"]
        C["area = distance"]
    end

Exam Tip: Do not confuse the two types of graph. Check the axis labels carefully. A flat line on a d–t graph means stationary, but a flat line on a v–t graph means constant velocity.

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Uniform and Non-Uniform Acceleration

• Uniform acceleration means the velocity changes by equal amounts in equal time intervals. The v–t graph is a straight line.
• Non-uniform acceleration means the rate of change of velocity is not constant. The v–t graph is a curve.

For non-uniform motion, the instantaneous acceleration at a point is found by drawing a tangent to the curve and calculating its gradient.

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Summary

• Speed is scalar (magnitude only); velocity is vector (magnitude + direction).
• Acceleration = rate of change of velocity: $a = (v - u) / t$a=(v−u)/t.
• Negative acceleration = deceleration (the object is slowing down).
• On a velocity–time graph: gradient = acceleration, area under the graph = distance travelled.
• A horizontal line on a v–t graph means constant velocity (zero acceleration).
• Split complex v–t graphs into rectangles and triangles to find total distance.
• Uniform acceleration gives a straight line; non-uniform gives a curve.

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Extended Worked Examples

Example 4: Rearranging the acceleration equation

A train accelerates uniformly. It reaches 30 m/s from 12 m/s while covering 630 m. Find the acceleration.

• Known: u = 12 m/s, v = 30 m/s, s = 630 m.
• Because time is not given, use $v^2 = u^2 + 2as$v2=u2+2as (introduced formally in the next lesson, but provided on the Edexcel formula sheet).
• Rearrange: $a = (v^2 - u^2)/(2s)$a=(v2−u2)/(2s).
• Substitute: $a = (900 - 144) / 1260 = 756 / 1260 = 0.60$a=(900−144)/1260=756/1260=0.60 m/s².

Example 5: Deceleration with sign conventions

A cyclist travelling east at 10 m/s brakes uniformly and comes to rest in 5.0 s.