Acceleration

📖 In-Depth Theory

ACCELERATION is the rate of change of velocity.

EQUATIONS:

a = Δv ÷ t = (v − u) ÷ t

a = acceleration (m/s²)

v = final velocity (m/s)

u = initial velocity (m/s)

t = time (s)

Also: v² = u² + 2as (links velocity, acceleration and distance)

a is positive → SPEEDING UP (in direction of motion).

a is negative → SLOWING DOWN (deceleration).

EXAMPLE:

Car accelerates from 10 m/s to 30 m/s in 5 s:

a = (30 − 10) ÷ 5 = 20 ÷ 5 = 4 m/s²

NOTE: acceleration can occur even at constant speed if DIRECTION changes (circular motion).

Velocity–Time Graphs

A VELOCITY–TIME GRAPH shows velocity (y-axis) against time (x-axis).

GRADIENT = ACCELERATION:

Steep positive gradient → large acceleration.

Flat (horizontal) → constant velocity (zero acceleration).

Negative gradient → deceleration.

AREA UNDER THE GRAPH = DISTANCE TRAVELLED:

For constant acceleration (straight line): area = ½ × base × height (triangle) + rectangle.

For horizontal section: area = base × height (rectangle).

SHAPES:

Horizontal line → constant velocity.

Straight line up → constant acceleration.

Straight line down → constant deceleration.

Curved → changing acceleration.

Using v² = u² + 2as

The equation v² = u² + 2as is useful when TIME is not given.

EXAMPLE:

A car starts from rest (u = 0) and accelerates at 3 m/s² over 75 m. Find the final speed.

v² = 0² + 2 × 3 × 75 = 450

v = √450 ≈ 21.2 m/s

EXAMPLE 2:

Car moving at 20 m/s decelerates at 4 m/s² to a stop. Find the braking distance.

v = 0, u = 20 m/s, a = −4 m/s²

0 = 400 + 2 × (−4) × s

8s = 400

s = 50 m

CALCULATING DISTANCE FROM v–t GRAPH:

Distance = area under the v–t graph.

Use triangle area (½bh) for constant acceleration from rest.

Use trapezium area for constant acceleration with initial velocity.

⚠️ Common Mistake

Gradient of a v–t graph = ACCELERATION (not distance or speed). Area under a v–t graph = DISTANCE. This is the opposite of the d–t graph where gradient = speed. Don't confuse the two graphs.

📐 Variables

aAcceleration (a) is measured in m/s² (m/s²)

vFinal velocity (v) is measured in m/s (m/s)

uInitial velocity (u) is measured in m/s (m/s)

tTime (t) is measured in seconds (s)

sDistance (s) is measured in metres (m)

📐 Key Equations

a = (v − u) ÷ t

v² = u² + 2as

📌 Key Note

a = (v−u)/t. v² = u² + 2as. Units: m/s². Positive a = speeding up; negative a = decelerating. v–t graph: gradient = acceleration; area = distance. Horizontal v–t = constant velocity. Steep v–t = large acceleration.

🎯 Matching Activity — Acceleration Calculations

Match each scenario to the correct acceleration or distance. — drag the symbols on the right to match the component names on the left.

4 m/s²

Drop here

−2 m/s²

Drop here

Area = distance

Drop here

Gradient = acceleration

Drop here

The gradient of a velocity–time graph gives acceleration

Velocity increases from 10 m/s to 30 m/s in 5 s — a = 20÷5

The area under a velocity–time graph gives distance travelled

Car decelerates from 20 m/s to 0 in 10 s — a = −20÷10

⚽ FIFA Worked Examples

Acceleration

A train accelerates from rest to 50 m/s in 25 s. Calculate the acceleration.

F

a = (v − u) ÷ t

I

v = 50 m/s, u = 0 m/s, t = 25 s

F

a = (50 − 0) ÷ 25 = 50 ÷ 25

A

a = 2 m/s²

⭐ Higher Tier Only

Use the equations of motion quantitatively including v² = u² + 2as. Estimate the speed, acceleration or distance for situations including everyday transport. Interpret velocity-time graphs to find: acceleration from gradient, distance from area under the graph (trapezium or triangle calculation).

🎯 Test Yourself

Question 1 of 2

1. A car decelerates from 24 m/s to 6 m/s in 9 s. What is the acceleration?

2. A v–t graph shows a straight horizontal line at 12 m/s for 6 seconds. What is the distance covered?

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