1.3 - Acceleration

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.

The formula for acceleration

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 ($v-u$).

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.

Final velocity and average velocity

When acceleration is constant, two further relationships link the quantities of motion.

Final velocity

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)
• $t$ = Time taken, in seconds (s)

Average velocity

$$\text{Average velocity}=\frac{u+v}{2}$$

This formula for average velocity only applies when acceleration is constant.

Worked example - Final and average velocity

A cyclist starts at 4 m/s and accelerates uniformly at 2 m/s² for 6 s.

Free fall and the acceleration of free fall

• In the absence of air resistance, all objects fall at the same rate regardless of their mass.
• An object dropped from rest speeds up as it falls - its speed increases by about 10 m/s every second.
• The steady acceleration of an object falling freely from rest is called the acceleration of free fall, g. Near the Earth's surface its value is about 10 m/s².