Acceleration: how quickly the velocity changes

Definition

Acceleration

The change in velocity per unit time.

\frac{\Delta v}{\Delta t}

where $Δv\Delta v$ is the change in velocity in $m/s\text{m/s}$ and $Δt\Delta t$ is the time taken in $s\text{s}$ ; the unit of acceleration is $m/s2\text{m/s}^2$ .

The symbol $Δ\Delta$ means "change in", and a change is always final value minus initial value, so $\dfrac{v - u}{t}$ .
The unit follows from the equation: a velocity in $m/s\text{m/s}$ divided by a time in $s\text{s}$ gives metres per second, per second, written $m/s2\text{m/s}^2$ .
An acceleration of $m/s23\ \text{m/s}^2$ means the velocity changes by $m/s3\ \text{m/s}$ during every second.

Practical

Investigation: measuring speed and acceleration with a tickertape timer

Aim: to use a tickertape timer to find the speed of a trolley and to identify and calculate its acceleration.
Apparatus: tickertape timer and power supply, paper tape, trolley, a sloping runway, and a ruler.
Method:
- Attach the tape to the trolley and thread it through the timer.
- Switch the timer on and release the trolley down the sloping runway.
- The timer prints 50 dots per second, so a ten-tick strip lasts $10 \times \tfrac{1}{50} = 0.2\ \text{s}$ ; measure a strip's length to get the average speed over it from $\text{length} / 0.2\ \text{s}$ .
- Compare successive ten-tick strips and find the acceleration from $\Delta v / \Delta t$ .
Measure and plot:
- Measure the length of each ten-tick strip, which gives the speed over it.
- Build a tape chart of successive strips to show how the speed changes.
Result: equal strip lengths mean constant speed; strips that lengthen evenly mean constant acceleration, with $\Delta v / \Delta t$ .
Watch out:
- The dots bunch together at the crowded start, so ignore that region.
- Average over several ten-tick strips rather than trusting a single reading.
Safety: stop the trolley before it runs off the end of the runway.

Constant and changing acceleration

On a speed-time graph, how steeply the line rises shows how quickly the speed is changing.
A horizontal line means the speed is not changing: zero acceleration (constant speed).
A straight sloping line means the speed changes by the same amount every second: constant (uniform) acceleration; the steeper the line, the larger the acceleration.
A curved line means changing (non-uniform) acceleration, a curve that steepens shows an increasing acceleration, one that flattens shows a decreasing acceleration.
From data, compare the change in speed over equal time intervals: equal changes mean constant acceleration, unequal changes mean changing acceleration.

Example

Two vehicles have their speed recorded every $s1.0\ \text{s}$ .
Vehicle A: $m/s0,\ 4,\ 8,\ 12,\ 16\ \text{m/s}$ . The speed rises by $m/s4\ \text{m/s}$ every second, so the acceleration is constant at $m/s24\ \text{m/s}^2$ and the graph is a straight sloping line.
Vehicle B: $m/s0,\ 6,\ 10,\ 12,\ 13\ \text{m/s}$ . The rises are $6$ , $4$ , $2$ and $m/s1\ \text{m/s}$ , so the acceleration is changing, it is decreasing, and the graph is a curve that flattens out.
Vehicle B is speeding up the whole time, even though its acceleration is falling.

Deceleration is a negative acceleration

Definition

Deceleration

A negative acceleration: the object is slowing down, so its final velocity is less than its initial velocity.

There is no separate equation: use $\Delta v / \Delta t$ , take the direction of travel as positive, and because $v$ is less than $u$ , $Δv\Delta v$ comes out negative and so does $a$ .
Keep the minus sign in your working; it carries the physics.

Common Mistake

A negative acceleration does not mean moving backwards; it means slowing down while still travelling forwards.
Do not write "a deceleration of $m/s2-2\ \text{m/s}^2$ ", which says it is speeding up. Either give the acceleration as $m/s2-2\ \text{m/s}^2$ , or call it a deceleration of $m/s22\ \text{m/s}^2$ .
Constant acceleration is not constant speed: on a speed-time graph a straight slope is constant acceleration, a horizontal line is constant speed and zero acceleration.
Do not read a speed-time graph as a distance-time graph: gradient of distance-time is speed, gradient of speed-time is acceleration, area under speed-time is distance.

Three velocity-time graphs: a horizontal line for constant velocity, a straight line with a positive gradient for uniform acceleration, and a straight line with a negative gradient for uniform deceleration.

Exam technique

Write $\Delta v / \Delta t$ from recall before substituting, and show $v - u$ as its own step, so staged marks still score if the arithmetic slips.
Always give the unit $m/s2\text{m/s}^2$ , and convert a time given in minutes to seconds first.
When asked to judge "from the shape of the graph", name the shape and state the conclusion ("a straight slope, so constant acceleration"); no calculation is wanted.

Self review

Define acceleration, and give its equation and unit.
What does $Δv\Delta v$ stand for, and how do you work it out?
On a speed-time graph, how do you tell constant acceleration from changing acceleration?
Why does a deceleration come out as a negative acceleration?
Does a negative acceleration mean the object is moving backwards?

Acceleration: how quickly the velocity changes

Definition

Acceleration

The change in velocity per unit time.

\frac{\Delta v}{\Delta t}

where $Δv\Delta v$ is the change in velocity in $m/s\text{m/s}$ and $Δt\Delta t$ is the time taken in $s\text{s}$ ; the unit of acceleration is $m/s2\text{m/s}^2$ .

The symbol $Δ\Delta$ means "change in", and a change is always final value minus initial value, so $\dfrac{v - u}{t}$ .
The unit follows from the equation: a velocity in $m/s\text{m/s}$ divided by a time in $s\text{s}$ gives metres per second, per second, written $m/s2\text{m/s}^2$ .
An acceleration of $m/s23\ \text{m/s}^2$ means the velocity changes by $m/s3\ \text{m/s}$ during every second.

Practical

Investigation: measuring speed and acceleration with a tickertape timer

Aim: to use a tickertape timer to find the speed of a trolley and to identify and calculate its acceleration.
Apparatus: tickertape timer and power supply, paper tape, trolley, a sloping runway, and a ruler.
Method:
- Attach the tape to the trolley and thread it through the timer.
- Switch the timer on and release the trolley down the sloping runway.
- The timer prints 50 dots per second, so a ten-tick strip lasts $10 \times \tfrac{1}{50} = 0.2\ \text{s}$ ; measure a strip's length to get the average speed over it from $\text{length} / 0.2\ \text{s}$ .
- Compare successive ten-tick strips and find the acceleration from $\Delta v / \Delta t$ .
Measure and plot:
- Measure the length of each ten-tick strip, which gives the speed over it.
- Build a tape chart of successive strips to show how the speed changes.
Result: equal strip lengths mean constant speed; strips that lengthen evenly mean constant acceleration, with $\Delta v / \Delta t$ .
Watch out:
- The dots bunch together at the crowded start, so ignore that region.
- Average over several ten-tick strips rather than trusting a single reading.
Safety: stop the trolley before it runs off the end of the runway.

Constant and changing acceleration

On a speed-time graph, how steeply the line rises shows how quickly the speed is changing.
A horizontal line means the speed is not changing: zero acceleration (constant speed).
A straight sloping line means the speed changes by the same amount every second: constant (uniform) acceleration; the steeper the line, the larger the acceleration.
A curved line means changing (non-uniform) acceleration, a curve that steepens shows an increasing acceleration, one that flattens shows a decreasing acceleration.
From data, compare the change in speed over equal time intervals: equal changes mean constant acceleration, unequal changes mean changing acceleration.

Example

Two vehicles have their speed recorded every $s1.0\ \text{s}$ .
Vehicle A: $m/s0,\ 4,\ 8,\ 12,\ 16\ \text{m/s}$ . The speed rises by $m/s4\ \text{m/s}$ every second, so the acceleration is constant at $m/s24\ \text{m/s}^2$ and the graph is a straight sloping line.
Vehicle B: $m/s0,\ 6,\ 10,\ 12,\ 13\ \text{m/s}$ . The rises are $6$ , $4$ , $2$ and $m/s1\ \text{m/s}$ , so the acceleration is changing, it is decreasing, and the graph is a curve that flattens out.
Vehicle B is speeding up the whole time, even though its acceleration is falling.

Deceleration is a negative acceleration

Definition

Deceleration

A negative acceleration: the object is slowing down, so its final velocity is less than its initial velocity.

There is no separate equation: use $\Delta v / \Delta t$ , take the direction of travel as positive, and because $v$ is less than $u$ , $Δv\Delta v$ comes out negative and so does $a$ .
Keep the minus sign in your working; it carries the physics.

Common Mistake

A negative acceleration does not mean moving backwards; it means slowing down while still travelling forwards.
Do not write "a deceleration of $m/s2-2\ \text{m/s}^2$ ", which says it is speeding up. Either give the acceleration as $m/s2-2\ \text{m/s}^2$ , or call it a deceleration of $m/s22\ \text{m/s}^2$ .
Constant acceleration is not constant speed: on a speed-time graph a straight slope is constant acceleration, a horizontal line is constant speed and zero acceleration.
Do not read a speed-time graph as a distance-time graph: gradient of distance-time is speed, gradient of speed-time is acceleration, area under speed-time is distance.

Exam technique

Write $\Delta v / \Delta t$ from recall before substituting, and show $v - u$ as its own step, so staged marks still score if the arithmetic slips.
Always give the unit $m/s2\text{m/s}^2$ , and convert a time given in minutes to seconds first.
When asked to judge "from the shape of the graph", name the shape and state the conclusion ("a straight slope, so constant acceleration"); no calculation is wanted.

Self review

Define acceleration, and give its equation and unit.
What does $Δv\Delta v$ stand for, and how do you work it out?
On a speed-time graph, how do you tell constant acceleration from changing acceleration?
Why does a deceleration come out as a negative acceleration?
Does a negative acceleration mean the object is moving backwards?

1.2.4 Acceleration and deceleration

Acceleration: how quickly the velocity changes

Investigation: measuring speed and acceleration with a tickertape timer

Constant and changing acceleration

Deceleration is a negative acceleration

Review 1.2.4 Acceleration and deceleration by teaching Genie

1.2.4 Acceleration and deceleration

Acceleration: how quickly the velocity changes

Investigation: measuring speed and acceleration with a tickertape timer

Constant and changing acceleration

Deceleration is a negative acceleration

Review 1.2.4 Acceleration and deceleration by teaching Genie

1.2.4 Acceleration and deceleration

Acceleration: how quickly the velocity changes

Investigation: measuring speed and acceleration with a tickertape timer

Constant and changing acceleration

Deceleration is a negative acceleration

Review 1.2.4 Acceleration and deceleration by teaching Genie

1.2.4 Acceleration and deceleration Revision Guide

1.2.4 Acceleration and deceleration

Acceleration: how quickly the velocity changes

Investigation: measuring speed and acceleration with a tickertape timer

Constant and changing acceleration

Deceleration is a negative acceleration

Review 1.2.4 Acceleration and deceleration by teaching Genie

1.2.4 Acceleration and deceleration Revision Guide