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.

Velocity-Time Graphs

A velocity-time graph shows how an object's velocity changes over time.

Velocity-time graphs allow us to determine much more than how fast an object is moving. We can use them to identify:

• velocity at a particular time
• direction of motion
• constant velocity
• acceleration
• deceleration
• changes in direction

The slope (gradient) of a velocity-time graph represents the object's acceleration.

 

 

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Reading a Velocity-Time Graph

A velocity-time graph has two axes.

Horizontal Axis

The horizontal or x-axis represents:

Time (s)

Vertical Axis

The vertical or y-axis represents:

Velocity (m/s)

Unlike a speed-time graph, the vertical axis can contain both positive and negative values.

This is because velocity includes direction.

Velocity Includes Direction

Velocity describes both:

• how fast an object is moving
• the direction in which it is moving

We usually choose one direction to be positive.

For example:

Right = positive

Left = negative

An object travelling at:

+8 m/s

could therefore be moving right at 8 m/s.

An object travelling at:

−8 m/s

would be moving at the same speed but in the opposite direction.

The negative sign does not mean the object is moving slowly.

It tells us about its direction.

Positive and Negative Velocity

On a velocity-time graph:

Above the time axis → Positive velocity

Below the time axis → Negative velocity

On the time axis → Zero velocity

 

 

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For example:

Constant Velocity

A horizontal line on a velocity-time graph represents constant velocity.

For example, suppose a car travels at:

12 m/s for 5 seconds

Its velocity does not change.

Therefore:

Acceleration = 0 m/s²

The velocity-time graph would show a horizontal line at +12 m/s.

Constant velocity

An object travels at a constant velocity of 12 m/s.

A horizontal line therefore means:

Constant velocity → Zero acceleration

Acceleration and Slope

The slope of a velocity-time graph represents acceleration.

The equation is:

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

where:

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

A steeper slope represents a larger magnitude of acceleration.

Calculating Acceleration from the Graph

Suppose an object's velocity increases from:

4 m/s to 16 m/s

during:

6 seconds

Step 1 – Find the change in velocity

Δv = 16 - 4 = 12 m/s

Step 2 – Divide by the time

\( a = \frac{12}{6} = 2 m/s^2 \)

The object accelerates at 2 m/s².

Positive Acceleration

Consider an object whose velocity changes as follows:

The line slopes upward.

Acceleration:

\( a = \frac{10 - 0}{5} = 2m/s^2 \)

The object has positive acceleration.

Negative Acceleration

Now suppose the velocity changes from:

20 m/s → 0 m/s

over 5 seconds.

\( a = \frac{0 - 20}{5} = -4m/s^2 \)

The negative acceleration means the velocity is changing in the negative direction.

In this particular example, the object has positive velocity but is slowing down, so we can also say it is decelerating.

 

 

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Deceleration

Deceleration means that an object's speed is decreasing.

For an object moving in the positive direction, deceleration appears as a line sloping downward toward zero.

For example:

15 m/s → 10 m/s → 5 m/s → 0 m/s

The object becomes progressively slower until it stops.

However, we need to be careful:

Negative acceleration does not always mean deceleration.

An object moving in the negative direction can have negative acceleration and actually speed up.

Acceleration vs Deceleration

The easiest way to determine whether an object is speeding up or slowing down is to look at the magnitude of its velocity.

Moving Away from Zero

Speed is increasing.

Object is speeding up.

Moving Toward Zero

Speed is decreasing.

Object is slowing down.

For example:

−2 m/s → −4 m/s → −6 m/s

The values are becoming more negative, but the object's speed is increasing:

2 m/s → 4 m/s → 6 m/s

So the object is speeding up in the negative direction.

Crossing the Time Axis

One of the most important features of a velocity-time graph occurs when the line crosses:

v = 0

At that instant, the object has zero velocity.

If the velocity then changes sign, the object has changed direction.

For example:

+6 m/s → +3 m/s → 0 m/s → −3 m/s → −6 m/s

The object:

1. moves in the positive direction
2. slows down
3. momentarily stops
4. reverses direction
5. speeds up in the negative direction

 

 

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This is a major difference between speed-time graphs and velocity-time graphs.

A speed-time graph cannot have negative speed.

A velocity-time graph can have negative velocity.

Interpreting Different Sections

A velocity-time graph may contain several different sections.

Imagine the following journey:

Section A

Velocity increases from:

0 → +10 m/s

The object is accelerating in the positive direction.

Section B

Velocity remains:

+10 m/s

The object travels at constant positive velocity.

Section C

Velocity decreases:

+10 → 0 m/s

The object decelerates to a stop.

Section D

Velocity becomes:

0 → −5 m/s

The object changes direction and speeds up in the negative direction.

Section E

Velocity remains:

−5 m/s

The object moves at constant velocity in the negative direction.

A Complete Journey

Consider this example:

We can analyse each section.

0–2 seconds

Velocity:

0 → +6 m/s

The object accelerates.

\( a = \frac{6 - 0}{2} = 3m/s^2 \)

2–4 seconds

Velocity remains:

+6 m/s

The object moves at constant positive velocity.

a = 0 m/s²

4–6 seconds

Velocity:

+6 → 0 m/s

\( a = \frac{0 - 6}{2} = -3m/s^2 \)

The object slows to a stop.

6–8 seconds

Velocity:

0 → −4 m/s

\( a = \frac{-4 - 0}{2} = -2m/s^2 \)

The object accelerates in the negative direction.

8–10 seconds

Velocity remains:

−4 m/s

The object moves at a constant velocity in the negative direction.

a = 0 m/s²

Understanding the Slope

The slope tells us how rapidly velocity is changing.

Remember:

Slope = acceleration

This is one of the most important relationships when interpreting a velocity-time graph.

Constructing a Velocity-Time Graph

Suppose we are given the following description:

A cyclist starts from rest and accelerates uniformly to 8 m/s in 4 seconds. The cyclist travels at 8 m/s for another 3 seconds before slowing uniformly to rest over 2 seconds.

First, create a table.

Then:

1. put time on the x-axis
2. put velocity on the y-axis
3. choose an appropriate scale
4. plot the points
5. connect the points with straight lines

Analysing the Cyclist's Graph

0–4 seconds

The cyclist accelerates:

\( a = \frac{8 - 0}{4} = 2m/s^2 \)

4–7 seconds

Velocity remains constant at: 8m/s

Therefore:

a = 0 m/s²

7–9 seconds

The cyclist slows:

\( a = \frac{0 - 8}{2} = -4m/s^2 \)

Notice that the final section is steeper than the first.

Therefore, the magnitude of the cyclist's deceleration is greater than the magnitude of the initial acceleration.

Velocity-Time Graphs and Real Motion

Velocity-time graphs can describe many real situations.

For example, a car approaching traffic lights might:

Accelerate → Constant velocity → Decelerate → Stop

An elevator might:

Accelerate upward → Constant upward velocity → Decelerate → Stop

Then later:

Accelerate downward → Constant negative velocity → Decelerate → Stop

 

 

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A Good Strategy for Reading Any Velocity-Time Graph

When you see a velocity-time graph, work through it systematically.

First: Look at whether the graph is above or below zero.

This tells you the direction of motion.

Second: Look at the slope.

This tells you the acceleration.

Third: Look for horizontal sections.

These represent constant velocity.

Fourth: Look for places where the graph reaches or crosses zero.

These may represent the object stopping or changing direction.

Finally: Calculate slopes when numerical acceleration values are required.

Common Misconception

A downward-sloping line does not always mean the object is slowing down.

Consider:

0 → −5 → −10 m/s

The graph slopes downward, so acceleration is negative.

But the object's speed changes:

0 → 5 → 10 m/s

The object is actually speeding up in the negative direction.

Therefore:

Negative acceleration ≠ always slowing down

Instead, compare the direction of velocity and acceleration.

Did You Know?

Velocity-time graphs can also tell us an object's displacement.

The displacement during a time interval is equal to the signed area between the graph and the time axis.

Areas above the axis represent displacement in the positive direction.

Areas below the axis represent displacement in the negative direction.

This makes velocity-time graphs especially powerful: the slope tells us acceleration, while the area tells us displacement.

Key Terms

Velocity-time graph – A graph showing how velocity changes with time.

Velocity – Speed in a specified direction.

Acceleration – Rate of change of velocity.

Deceleration – A decrease in speed.

Slope – The steepness of a graph; on a velocity-time graph it represents acceleration.

Constant velocity – Motion with unchanged velocity.

Positive velocity – Motion in the chosen positive direction.

Negative velocity – Motion in the direction opposite to the chosen positive direction.

Zero velocity – The object is stationary at that instant.

Key Takeaways

• A velocity-time graph shows how velocity changes with time.
• Time is plotted on the x-axis and velocity on the y-axis.
• The slope of a velocity-time graph represents acceleration.
• Acceleration can be calculated using \( a = \frac{ \Delta v }{ \Delta t } = \frac{v_f - v_i}{t} \).
• A horizontal line represents constant velocity and zero acceleration.
• An upward slope represents positive acceleration.
• A downward slope represents negative acceleration.
• Velocity above the time axis is positive.
• Velocity below the time axis is negative.
• Negative velocity means movement in the opposite direction, not negative speed.
• If the graph crosses the time axis, the object may be changing direction.
• Deceleration means speed is decreasing, so negative acceleration does not always mean deceleration.
• Velocity-time graphs can be constructed from motion descriptions or numerical data.
• The slope gives acceleration, while the signed area under the graph gives displacement.

Acceleration in the Real World

Acceleration is not just something we calculate in physics problems. It occurs whenever an object's velocity changes, making it important in transportation, sports, engineering, nature, and safety.

Remember:

Acceleration is the rate of change of velocity.

An object accelerates whenever it:

• speeds up
• slows down
• changes direction
• changes both speed and direction

This means acceleration occurs in far more situations than simply a car pressing its accelerator.

 

 

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Acceleration in Transportation

Vehicles constantly experience acceleration.

Consider a car travelling through a city.

The car may:

Start from rest → Speed up → Travel at constant velocity → Brake → Turn → Stop

Acceleration occurs during every stage except when the car travels at a constant velocity in a straight line.

Examples include:

• cars leaving traffic lights
• buses braking at stops
• aircraft taking off
• trains leaving stations
• bicycles turning corners
• roller coasters changing speed and direction

Vehicle Performance

Acceleration is an important measure of vehicle performance.

A car that changes velocity from: 0 m/s to 25 m/s in 5 seconds has an average acceleration of:

\( a = \frac{ \Delta v }{ \Delta t } = \frac{\ 25 - 0}{5} = 5m/s^2 \)

This means the car's velocity increases by an average of 5 m/s every second.

 

 

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Comparing Vehicle Acceleration

Suppose two cars accelerate from rest.

Car A

Reaches 20 m/s in 4 seconds.

\( a = \frac{20}{4} = 5m/s^2 \)

Car B

Reaches 20 m/s in 8 seconds.

\( a = \frac{20}{8} = 2.55m/s^2 \)

Car A has the greater average acceleration.

It changes its velocity more rapidly.

This is why acceleration times such as 0–100 km/h are often used when comparing vehicle performance.

Acceleration and Braking

Braking also involves acceleration.

Suppose a car is travelling at: 24 m/s and stops in 6 s.

Its acceleration is:

\( a = \frac{0 - 24}{6} = -4m/s^2 \)

The negative sign indicates that the acceleration acts opposite to the chosen positive direction of motion.

Because the car's speed is decreasing, we can also describe this as deceleration.

Acceleration While Turning

A vehicle can accelerate even when its speed remains constant.

Imagine a car travelling around a roundabout at a constant: 10m/s

Its speed remains the same, but its direction continuously changes.

Since velocity includes direction:

Changing direction → changing velocity

Therefore: The car is accelerating

 

 

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This inward acceleration during circular motion is called centripetal acceleration.

Acceleration in Sports

Sports provide many excellent examples of acceleration.

An athlete accelerates whenever they change their speed or direction.

Examples include:

• a sprinter leaving the starting blocks
• a football player changing direction
• a cyclist accelerating out of a corner
• a tennis ball changing velocity after being struck
• a basketball falling toward the floor
• a diver accelerating toward the water
• a skier travelling around a turn

 

 

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A Sprinter

A sprinter starts from rest and reaches: 10 m/s after 2.5 s

Average acceleration:

\( a = \frac{10 - 0}{2.5} = 4m/s^2 \)

The sprinter's velocity increases rapidly during the beginning of the race.

Later, the runner may reach approximately constant speed.

At that stage: a ≈ 0

provided the runner is moving in a straight line at approximately constant speed.

Changing Direction in Sports

Imagine a football player running east at: 6 m/s

The player suddenly turns and begins running north at the same speed.

Has the player accelerated?

Yes.

The speed is still: 6 m/s

but the direction has changed.

Therefore the velocity changed.

This demonstrates again that:

Acceleration does not require a change in speed.

Acceleration Data

Real motion can be analysed using data collected from:

• speed sensors
• GPS devices
• motion detectors
• accelerometers
• smartphones
• vehicle computers

Consider the following measurements from a cyclist:

We can use the data to analyse the cyclist's motion.

Analysing the Cyclist's Data

0–4 seconds

Velocity increases: 2 → 10 m/s

Average acceleration:

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

The cyclist is speeding up.

4–6 seconds

Velocity remains: 10 m/s

Therefore: a = 0

The cyclist moves at constant velocity.

6–10 seconds

Velocity changes: 10 → 2 m/s

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

The cyclist is slowing down.

Acceleration and Vehicle Safety

Acceleration is extremely important when considering vehicle collisions.

During a collision, a vehicle may change from a large velocity to zero in a very short time.

For example, suppose a vehicle travelling at: 20 m/s stops in 0.10s

Average acceleration:

\( a = \frac{0 - 20}{0.10} = -200m/s^2 \)

This is an extremely large acceleration magnitude.

Large accelerations during collisions can produce large forces on passengers.

Why Increasing Stopping Time Helps

From:

\( a = \frac{ \Delta v }{ \Delta t } \)

we can see that for the same change in velocity:

Larger stopping time → Smaller acceleration magnitude

For example, consider the same velocity change:

20 m/s → 0

Stopping in 0.10 s

a = -200 m/s²

Stopping in 0.50 s

a = -40 m/s²

Increasing the stopping time dramatically reduces the magnitude of the acceleration.

This principle is central to many vehicle safety systems.

Seat Belts

When a vehicle suddenly stops, passengers tend to continue moving because of their inertia.

A seat belt restrains the passenger and helps bring the passenger to rest with the vehicle.

Modern seat belts can also work with other safety systems to manage the forces acting on the body during a collision.

 

 

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Without a seat belt, the passenger may continue moving until striking the:

• dashboard
• steering wheel
• windscreen
• seat in front

The seat belt helps control this rapid change in velocity.

Airbags

Airbags provide another way of reducing injury during a collision.

The airbag inflates rapidly and provides a surface that helps bring the passenger to rest over a greater time and distance than a hard dashboard or steering wheel would.

For the same change in velocity:

Longer stopping time → Smaller acceleration magnitude

This can reduce the force acting on the passenger.

 

 

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Airbags are designed to work with seat belts, not replace them.

Crumple Zones

Cars also contain crumple zones.

These parts of the vehicle are designed to deform during a collision.

The deformation:

• absorbs and redirects some energy
• increases the time over which the vehicle changes velocity
• reduces the magnitude of the acceleration experienced by the passenger compartment compared with a more abrupt stop

This demonstrates how understanding acceleration can directly influence engineering and safety design.

Acceleration in Nature

Acceleration also occurs throughout nature.

Examples include:

• falling objects
• rain falling toward Earth
• rocks rolling down hills
• animals running and turning
• ocean waves moving objects
• planets orbiting stars
• moons orbiting planets
• meteors falling through atmospheres

One of the most important natural examples is free fall.

Free Fall

An object is in free fall when gravity is the only significant force acting on it.

Near Earth's surface, the acceleration due to gravity is approximately:

g = 9.8 m/s²

directed downward.

In introductory calculations, this is sometimes rounded to:

g ≈ 10 m/s²

This means a freely falling object's downward velocity changes by approximately 9.8 m/s every second.

h(t) = h_o ​+ v_o​t − ½​gt²

Falling from Rest

Suppose a ball is dropped from rest and we ignore air resistance.

Its approximate downward velocities would be:

Time	Downward Velocity
0 s	0 m/s
1 s	9.8 m/s
2 s	19.6 m/s
3 s	29.4 m/s
4 s	39.2 m/s

The velocity increases by 9.8 m/s each second.

Therefore, its acceleration is 9.8 m/s² downward

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Does a Falling Object Keep Accelerating Forever?

The simple free-fall model assumes there is no air resistance.

Real objects falling through Earth's atmosphere experience another force:

drag, or air resistance.

Drag acts opposite to the direction of motion.

As a falling object becomes faster, air resistance generally becomes larger.

 

 

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Terminal Velocity

Consider a skydiver falling through the atmosphere.

Initially:

Weight > Air resistance

There is a downward net force.

Therefore the skydiver accelerates downward.

As speed increases:

Air resistance increases

Eventually: Air resistance = Weight

The forces are balanced.

Therefore: F_net = 0 and: a = 0

The skydiver continues falling at a constant velocity called terminal velocity.

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Terminal Velocity Does Not Mean Stopping

This is an important distinction.

At terminal velocity: a = 0 but v ≠ 0

The object is still moving.

Its velocity simply stops changing.

Therefore: Zero acceleration does not mean zero velocity.

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Opening a Parachute

When a skydiver opens a parachute, the surface area exposed to the air increases dramatically.

This produces much greater drag.

Immediately after the parachute opens: Drag > Weight

The net force acts upward while the skydiver is still moving downward.

Therefore, the skydiver's downward speed decreases.

Eventually, the forces balance again: Drag = Weight

The skydiver reaches a new, much lower terminal velocity.

 

 

 

 

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A Terminal-Velocity Journey

A skydiver's motion can be divided into several stages.

Stage 1 – Jump

Velocity is initially small.

Gravity produces a downward acceleration.

Stage 2 – Speed Increases

The skydiver becomes faster.

Air resistance increases.

Stage 3 – First Terminal Velocity

Drag equals weight. a = 0

Velocity becomes approximately constant.

Stage 4 – Parachute Opens

Drag suddenly becomes much larger.

The skydiver slows rapidly.

Stage 5 – New Terminal Velocity

Drag and weight become balanced again.

The skydiver continues downward at a much slower constant velocity.

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Solving Practical Acceleration Problems

Consider a motorcycle that increases its velocity from: 8 m/s to 26 m/s in 6 s

Step 1 – Find the velocity change

Δv = 26 - 8 = 18 m/s

Step 2 – Divide by time

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

Interpretation:

The motorcycle's velocity increases by an average of 3 m/s each second.

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Practical Problem: Braking

A cyclist travelling at 12 m/s brakes and stops in 3s.

Calculate the acceleration.

\( a = \frac{0 - 12}{3} = -4m/s^2 \)

The negative acceleration shows that the velocity is decreasing in the chosen positive direction.

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Practical Problem: Free Fall

A stone is dropped from rest.

Ignoring air resistance, estimate its velocity after 3 s.

Using:

v_f = v_i + at

we have:

v_i = 0

a = 9.8 m/s² downward

Therefore:

v_f = (0) + (9.8)(3) = 29.4 m/s downward

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Using Acceleration Data to Make Decisions

Acceleration data can help answer practical questions.

For example:

Which car accelerates faster?

Compare changes in velocity over equal times.

Which cyclist brakes more sharply?

Compare the magnitudes of their negative accelerations.

When has a falling object reached terminal velocity?

Look for the point when velocity becomes constant and acceleration becomes zero.

Why can airbags reduce injuries?

They help increase the stopping time and distance, reducing the magnitude of acceleration and the force experienced.

This shows how mathematical calculations can help us interpret real physical situations.

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Reading Acceleration from a Velocity-Time Graph

Acceleration can also be determined graphically.

On a velocity-time graph: slope = acceleration

A steep upward slope indicates a large positive acceleration.

A horizontal line indicates: a = 0

A downward slope indicates negative acceleration.

 

 

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This makes velocity-time graphs particularly useful when analysing experimental or real-world motion data.

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Acceleration and Forces

Acceleration is produced when there is a net force acting on an object.

Newton's Second Law describes this relationship:

F_net = ma

Therefore, for a particular mass:

Larger net force → Larger acceleration

This connects acceleration to:

• vehicle engines
• braking
• collisions
• falling objects
• sports
• rockets
• turning vehicles

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Common Misconceptions

Acceleration does not only mean speeding up.

Slowing down and changing direction are also forms of acceleration.

Negative acceleration does not automatically mean moving backward.

It describes the direction of acceleration relative to the chosen coordinate system.

A falling object does not necessarily accelerate forever.

Air resistance can eventually produce terminal velocity.

Terminal velocity is not zero velocity.

At terminal velocity: v = constant

and: a = 0

Seat belts and airbags do not prevent the passenger's velocity from changing.

Instead, they help manage how the passenger comes to rest and reduce the severity of the forces involved.

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Did You Know?

Accelerometers are built into many smartphones.

They can detect changes in motion and orientation and contribute to features such as:

• screen rotation
• motion tracking
• fitness measurements
• gaming controls
• vehicle-motion detection

The same fundamental quantity studied in classroom physics—acceleration—is being measured continuously by devices people carry every day.

 

 

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Key Terms

Acceleration – Rate of change of velocity.

Deceleration – A decrease in speed.

Free fall – Motion in which gravity is the only significant force acting on an object.

Gravity – The attractive force between masses.

Acceleration due to gravity (g) – Approximately 9.8 m/s² downward near Earth's surface.

Air resistance – A drag force that opposes motion through air.

Terminal velocity – Constant falling velocity reached when drag balances weight.

Net force – The overall force resulting from all forces acting on an object.

Crumple zone – A vehicle structure designed to deform during a collision and help increase stopping time while managing energy.

Accelerometer – A sensor used to measure acceleration.

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Key Takeaways

• Acceleration occurs whenever velocity changes.
• Real-world acceleration occurs in transportation, sports, nature, and technology.
• Vehicles accelerate when they speed up, slow down, or turn.
• Acceleration data can be calculated using: \( a = \frac{ \Delta v }{ \Delta t } \)
• Vehicle performance can be compared using acceleration measurements.
• Large changes in velocity over very short times can produce large acceleration magnitudes.
• Seat belts, airbags, and crumple zones help manage rapid changes in velocity during collisions.
• Increasing stopping time can reduce the magnitude of acceleration and therefore help reduce forces on passengers.
• Athletes accelerate when they change speed or direction.
• Objects in free fall near Earth's surface accelerate at approximately: 9.8 m/s² downward when air resistance is neglected.
• Real falling objects experience air resistance.
• Terminal velocity occurs when drag balances weight.
• At terminal velocity, acceleration is zero, but the object is still moving.
• Opening a parachute increases drag and produces a much lower terminal velocity.
• Acceleration can be analysed using calculations, data tables, and velocity-time graphs.
• Understanding acceleration helps us solve practical problems involving vehicles, collisions, sports, free fall, and falling objects.