Physics

1.4 - Acceleration

8 min

The acceleration equation

Deceleration and uniform acceleration

Acceleration and distance

note 1 of 6

What acceleration is

Acceleration is how quickly velocity is changing - the change in velocity in a certain amount of time.
A change in velocity can be a change in speed, a change in direction, or both.
Acceleration is a vector, so it can be positive or negative.
Deceleration is negative acceleration: when something slows down, the change in velocity is negative.

note 2 of 6

The formula for acceleration

$a = \frac{v - u}{t}$

Where:

$a$ = Acceleration, in metres per second squared (m/s²)
$v$ = Final velocity, in metres per second (m/s)
$u$ = Initial velocity, in metres per second (m/s)
$t$ = Time taken, in seconds (s)

The change in velocity is the final velocity minus the initial velocity ( $v - u$ ).

note 3 of 6

Worked example - Time for a given acceleration

A cyclist accelerates at 1.5 m/s² from 3.0 m/s to 7.5 m/s. Find the time it takes to do this.

Step 1 - Rearrange the acceleration equation for time

$t = \frac{v - u}{a}$

Step 2 - Substitution and calculation

$t = \frac{7.5 - 3.0}{1.5} = 3.0 s$

note 4 of 6

Uniform acceleration and free fall

Uniform acceleration means a constant acceleration: the velocity changes by the same amount each second.
The acceleration of an object in free fall near the Earth's surface is uniform. It has the same value as the gravitational field strength, g.
Near the Earth's surface this value is taken as 10 m/s², so g = 10 N/kg.

note 5 of 6

Formula linking velocity, acceleration and distance

For an object moving with uniform (constant) acceleration, the final velocity, initial velocity, acceleration and distance are linked by this equation.

$v^{2} - u^{2} = 2 a x$

Where:

$v$ = Final velocity, in metres per second (m/s)
$u$ = Initial velocity, in metres per second (m/s)
$a$ = Acceleration, in metres per second squared (m/s²)
$x$ = Distance travelled, in metres (m)

This equation doesn't include time, so it's used when time isn't known.

note 6 of 6

Worked example - Final speed with v² = u² + 2ax

A car travelling at 21 m/s starts decelerating uniformly at 1.6 m/s² as it approaches a junction 100 m away. What will its speed be when it reaches the junction?

Step 1 - Identify the values

The acceleration is negative because the car is decelerating.

$\begin{matrix} u & = 21 m/s \\ a & = - 1.6 {m/s}^{2} \\ x & = 100 m \end{matrix}$

Step 2 - Choose the equation where v is the only unknown

$v^{2} = u^{2} + 2 a x$

Step 3 - Substitution and calculation

$\begin{matrix} v^{2} & = 21^{2} + (2 \times - 1.6 \times 100) \\ = 441 - 320 \\ = 121 \end{matrix}$

Step 4 - Take the square root

$v = \sqrt{121} = 11 m/s$

key terms

TermDefinition

AccelerationThe rate of change of velocity, equal to the change in velocity divided by the time taken.

DecelerationNegative acceleration, which happens when an object slows down.

Uniform accelerationA constant acceleration where the velocity changes by the same amount each second.

end of notes

Key points to remember

What acceleration is

Acceleration is how quickly velocity is changing - the change in velocity in a certain amount of time.

A change in velocity can be a change in speed, a change in direction, or both.

Acceleration is a vector, so it can be positive or negative.

Deceleration is negative acceleration: when something slows down, the change in velocity is negative.

Uniform acceleration and free fall

Uniform acceleration means a constant acceleration: the velocity changes by the same amount each second.

The acceleration of an object in free fall near the Earth's surface is uniform. It has the same value as the gravitational field strength, g.

Near the Earth's surface this value is taken as 10 m/s², so g = 10 N/kg.

Formula linking velocity, acceleration and distance

For an object moving with uniform (constant) acceleration, the final velocity, initial velocity, acceleration and distance are linked by this equation.

$v^{2} - u^{2} = 2 a x$

Where:

v

= Final velocity, in metres per second (m/s)

u

= Initial velocity, in metres per second (m/s)

a

= Acceleration, in metres per second squared (m/s²)

x

= Distance travelled, in metres (m)

This equation doesn't include time, so it's used when time isn't known.

Worked example - Final speed with v² = u² + 2ax

A car travelling at 21 m/s starts decelerating uniformly at 1.6 m/s² as it approaches a junction 100 m away. What will its speed be when it reaches the junction?

Step 1 - Identify the values

The acceleration is negative because the car is decelerating.

$\begin{matrix} u & = 21 m/s \\ a & = - 1.6 {m/s}^{2} \\ x & = 100 m \end{matrix}$

Step 2 - Choose the equation where v is the only unknown

$v^{2} = u^{2} + 2 a x$

Step 3 - Substitution and calculation

$\begin{matrix} v^{2} & = 21^{2} + (2 \times - 1.6 \times 100) \\ = 441 - 320 \\ = 121 \end{matrix}$

Step 4 - Take the square root

$v = \sqrt{121} = 11 m/s$

1.4 - Acceleration

What acceleration is

The formula for acceleration

Where:

Worked example - Time for a given acceleration

Step 1 - Rearrange the acceleration equation for time

Step 2 - Substitution and calculation

Uniform acceleration and free fall

Formula linking velocity, acceleration and distance

Where:

Worked example - Final speed with v2 = u2 + 2ax

Step 1 - Identify the values

Step 2 - Choose the equation where v is the only unknown

Step 3 - Substitution and calculation

Step 4 - Take the square root

Key points to remember

1.4 - Acceleration

What acceleration is

The formula for acceleration

Where:

Worked example - Time for a given acceleration

Step 1 - Rearrange the acceleration equation for time

Step 2 - Substitution and calculation

Uniform acceleration and free fall

Formula linking velocity, acceleration and distance

Where:

Worked example - Final speed with v2 = u2 + 2ax

Step 1 - Identify the values

Step 2 - Choose the equation where v is the only unknown

Step 3 - Substitution and calculation

Step 4 - Take the square root

Key points to remember

Worked example - Final speed with v² = u² + 2ax

Worked example - Final speed with v² = u² + 2ax