Acceleration

Definition of Acceleration

Acceleration is the rate of change of velocity with time. Since velocity is a vector (it has both size and direction), acceleration is also a vector quantity. This means acceleration can involve changes in speed, direction, or both.

Acceleration can be:

• Positive acceleration: when an object speeds up.
• Negative acceleration (also called deceleration): when an object slows down.

For example, when a car moves faster from rest, it has positive acceleration. When it brakes to stop, it has negative acceleration.

Calculating Acceleration

Acceleration is calculated by dividing the change in velocity by the time taken for that change:

$$\text{acceleration} = \frac{\text{change in velocity}}{\text{time taken}} = \frac{v - u}{t}$$

• $v$ = final velocity (m/s)
• $u$ = initial velocity (m/s)
• $t$ = time taken (s)

The unit of acceleration is metres per second squared ($\text{m/s}^2$), which means the velocity changes by a certain number of metres per second every second.

Velocity-time graphs are useful for finding acceleration. The acceleration is the gradient (slope) of the velocity-time graph:

• A straight line with a constant slope means uniform acceleration.
• A curved line means acceleration is changing (non-uniform acceleration).

For example, if a cyclist increases their velocity from 0 m/s to 10 m/s in 5 seconds, the acceleration is:

$$a = \frac{10 - 0}{5} = 2 \text{ m/s}^2$$

Types of Acceleration

Uniform acceleration means the velocity changes by the same amount every second. The acceleration is constant. For example, a car accelerating steadily from rest.

Non-uniform acceleration means the rate of change of velocity is not constant. The acceleration varies over time. For example, a car slowing down as it approaches a red light.

Deceleration is acceleration in the opposite direction to velocity, causing an object to slow down. It is represented by a negative acceleration value.

Acceleration in Motion

Acceleration affects both velocity and displacement. When acceleration is constant, the following equations of motion can be used to describe the motion of an object:

• $v = u + at$
• $s = ut + \frac{1}{2}at^2$
• $v^2 = u^2 + 2as$

• $u$ = initial velocity (m/s)
• $v$ = final velocity (m/s)
• $a$ = acceleration (m/s²)
• $t$ = time (s)
• $s$ = displacement (m)

These equations are useful for solving problems involving motion with constant acceleration, such as a car accelerating along a straight road or an object in free fall (ignoring air resistance).

For example, if a car starts from rest and accelerates uniformly at $3 \text{ m/s}^2$ for 4 seconds, its final velocity is:

$$v = u + at = 0 + (3 \times 4) = 12 \text{ m/s}$$

The displacement during this time is:

$$s = ut + \frac{1}{2}at^2 = 0 + \frac{1}{2} \times 3 \times 4^2 = 24 \text{ m}$$