What Is Acceleration?

Acceleration describes how quickly an object's velocity changes.

Because velocity includes both speed and direction, an object can accelerate by:

• speeding up
• slowing down
• changing direction
• changing both speed and direction

So acceleration is not simply "going faster."

A more complete definition is:

Acceleration is the rate of change of velocity.

 

 

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The Acceleration Equation

Acceleration can be calculated using:

\( a = \frac{ \Delta v }{ \Delta t } = \frac{v_f - v_i}{t} \)

where:

• a = acceleration in m/s²
• v_f = final velocity
• v_i = initial velocity
• t = time

The SI unit for acceleration is: m/s²

This means metres per second per second.

An acceleration of: 3 m/s²

means the velocity changes by 3 m/s every second.

Velocity and Acceleration Are Different

Velocity describes how fast an object is moving and in what direction.

Acceleration describes how quickly that velocity changes.

For example:

A car travelling east at a constant 20 m/s has velocity, but if its velocity is not changing, its acceleration is: 0 m/s²

Now suppose the car speeds up from: 20 m/s → 30 m/s

Its velocity has changed, so it is accelerating.

A useful comparison is:

Acceleration by Speeding Up

The easiest type of acceleration to recognise occurs when an object increases its speed.

Suppose a cyclist speeds up from: 4 m/s to 10 m/s in 3 seconds.

\( a = \frac{10 - 4}{3} = \frac{6}{3} = 2 m/s^2 \)

The cyclist's velocity increases by 2 m/s each second.

Acceleration by Slowing Down

An object is also accelerating when it slows down, because its velocity is changing.

Suppose a car slows from: 18 m/s to 6 m/s in 4 seconds.

\( a = \frac{6 - 18}{4} = \frac{-12}{4} = -3m/s^2 \)

The negative sign means the acceleration is in the negative direction relative to the chosen positive direction.

In this situation, because the car is moving forward but slowing down, the negative acceleration is also deceleration.

Acceleration by Changing Direction

An object can accelerate even if its speed stays constant.

This happens whenever its direction changes.

A car travelling around a circular bend at a constant speed is accelerating because its velocity is continuously changing direction.

 

 

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This is one of the most important ideas in motion:

Constant speed does not always mean zero acceleration.

If direction changes, velocity changes.

If velocity changes, the object accelerates.

Circular Motion

Consider a car travelling around a roundabout at a constant 10 m/s.

At one moment, it may be travelling north.

A few seconds later, it may be travelling east.

Its speed remains: 10 m/s but its direction has changed.

Therefore, its velocity has changed.

So the car is accelerating.

This is called centripetal acceleration, which is directed toward the centre of the circular path.

Positive Acceleration

Positive acceleration means acceleration in the chosen positive direction.

Suppose we define:

Right = positive

A car moving right changes velocity from +5 m/s to +15 m/s.

The acceleration is positive.

\( a = \frac{15 - 5}{5} = +2m/s^2 \)

The car is speeding up in the positive direction.

Negative Acceleration

Negative acceleration means acceleration in the chosen negative direction.

For example:

A car moving right slows from: +20 m/s to +10 m/s

Its acceleration is negative.

But negative acceleration does not always mean slowing down.

Suppose an object is moving left: -5 m/s and later moves at -12 m/s.

Its speed has increased from: 5 m/s → 12 m/s

The acceleration is negative, but the object is speeding up in the negative direction.

Positive and Negative Acceleration

The signs of velocity and acceleration tell us both direction and whether speed is changing.

A useful rule is:

Velocity and acceleration same sign → speeding up

Velocity and acceleration opposite signs → slowing down

Zero Acceleration

An object has zero acceleration when its velocity does not change.

This means both:

• speed remains constant
• direction remains constant

For example, a train travelling in a straight line at a constant 25 m/s has a = 0 m/s²

Even though the train is moving quickly, it is not accelerating because its velocity is constant.

Stationary Objects and Acceleration

A stationary object often has zero acceleration, but not always at every instant of motion.

For example, imagine throwing a ball straight upward.

At the highest point: v = 0

But gravity is still acting.

The acceleration is approximately: 9.8 m/s² downward.

So:

Zero velocity does not necessarily mean zero acceleration.

 

 

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Everyday Examples of Acceleration

Acceleration occurs constantly in everyday life.

Car Leaving Traffic Lights

The car speeds up from rest.

Velocity changes → acceleration

Bicycle Braking

The bicycle slows down.

Velocity changes → acceleration

Car Turning a Corner

The car changes direction.

Velocity changes → acceleration

Elevator Starting Upward

The elevator's velocity changes from zero to upward motion.

Acceleration occurs

Elevator Moving at Constant Speed

Velocity remains constant.

Acceleration = zero

Roller Coaster

Speed and direction change repeatedly.

Acceleration occurs frequently

 

 

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Acceleration in Free Fall

Objects falling near Earth's surface accelerate because of gravity.

Ignoring air resistance, the acceleration is approximately: g = 9.8 m/s² downward.

This means the downward velocity changes by about: 9.8 m/s every second.

For example, if an object is dropped from rest:

Its velocity is changing continuously, so it is accelerating.

A Graphical View of Acceleration

On a velocity-time graph, acceleration is represented by the slope.

A steep slope means a large acceleration.

A shallow slope means a smaller acceleration.

A horizontal line means: a = 0

 

 

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This gives us the important relationship:

slope of velocity-time graph = acceleration

Constant Acceleration

If velocity changes by the same amount every second, the object has constant acceleration.

For example:

The velocity increases by 3 m/s each second.

Therefore a = 3 m/s²

A velocity-time graph for this motion would be a straight sloping line.

Changing Acceleration

Acceleration itself can also change.

For example, when driving a car:

• pressing the accelerator gently may produce a small acceleration
• pressing it harder may produce a larger acceleration
• releasing it may reduce acceleration
• braking may produce acceleration in the opposite direction

If acceleration changes over time, the velocity-time graph may become curved rather than straight.

Worked Example

A runner increases velocity from 3 m/s to 9 m/s in 2 s.

Step 1 – Find the change in velocity

Δv = 9 - 3 = 6 m/s

Step 2 – Divide by time

\( a = \frac{6}{2} = 3 m/s^2 \)

The runner's velocity increases by 3 m/s each second.

Worked Example: Negative Velocity

Suppose east is positive.

A car changes velocity from -4 m/s to -12 m/s in 4 seconds.

\( a = \frac{-12 - (-4)}{4} = \frac{-8}{4} = -2 m/s^2 \)

The car has negative acceleration.

However, its speed increases from 4 m/s → 12 m/s

So it is speeding up in the negative direction.

Velocity vs Acceleration: A Useful Example

Imagine a car travelling along a road.

Situation A

Velocity: 20 m/s

Acceleration: 0

The car moves at constant velocity.

Situation B

Velocity: 20 m/s

Acceleration: +3 m/s²

The car is speeding up in the positive direction.

Situation C

Velocity: 20 m/s

Acceleration: -3m/s²

The car is slowing down.

The same velocity can therefore occur with very different accelerations.

Common Misconceptions

Acceleration does not always mean speeding up.

Slowing down and changing direction also involve acceleration.

Negative acceleration does not always mean slowing down.

It describes the direction of acceleration.

Zero acceleration does not mean zero velocity.

An object moving at constant velocity has zero acceleration.

Zero velocity does not always mean zero acceleration.

A ball at the highest point of its flight momentarily has zero velocity but still accelerates downward due to gravity.

Did You Know?

When a car travels around a circular track at a constant speed, the driver's speedometer may barely change.

However, the car is still continuously accelerating because its direction changes at every moment.

This is why passengers can feel pushed sideways when a vehicle turns, even if its speed stays constant.

Key Terms

Acceleration – The rate of change of velocity.

Velocity – Speed in a specified direction.

Positive acceleration – Acceleration in the chosen positive direction.

Negative acceleration – Acceleration in the chosen negative direction.

Zero acceleration – No change in velocity.

Deceleration – A decrease in speed.

Constant acceleration – Velocity changing by the same amount each second.

Centripetal acceleration – Acceleration directed toward the centre of circular motion.

Key Takeaways

• Acceleration is the rate of change of velocity.
• Acceleration is measured in m/s².
• An object accelerates when its speed changes, direction changes, or both.
• Velocity describes motion, while acceleration describes how velocity changes.
• Speeding up involves acceleration.
• Slowing down also involves acceleration.
• An object moving at constant speed can still accelerate if its direction changes.
• Positive and negative acceleration describe the direction of acceleration.
• Negative acceleration does not always mean slowing down.
• Zero acceleration means the object's velocity is constant.
• An object can have zero velocity but non-zero acceleration.
• Everyday examples include cars accelerating or braking, bicycles turning, elevators starting and stopping, free-falling objects, and roller coasters.