# Uniform motion and uniform acceleration

Uniform motion refers to movement at a constant velocity. In other words, an object is in uniform motion if it is moving at constant speed in a fixed direction.

Driving on a straight empty road at a constant speed is (approximately) uniform motion.

Uniform acceleration is an acceleration that is constant in magnitude over time.

If an object undergoes uniform acceleration with $$a=3 \text{ m/s}^2$$, then the velocity after one second is $$3 \text{ m/s}$$, after two seconds is $$6 \text{ m/s}$$ and after three seconds is $$9 \text{ m/s}$$.

A **free fall** (ignoring air resistance) is the most widely used example of uniform acceleration.

A constant change in direction (e.g. moving in a circle) at constant speed would also be uniform acceleration.

In the case of ** uniform acceleration**, several equations can be derived. We use the following symbols:

$$s=$$displacement; $$u=$$ initial velocity; $$v=$$ final velocity; $$a=$$ acceleration; $$t=$$ time taken.

These symbols give the equations their other commonly used name - SUVAT equations.

- From $$a=(v- u)/t$$: $$$\begin{gather} t=\frac{v-u}{a} \\ \Tblue{v=u+at} \end{gather}$$$
- From the average velocity $$\langle v\rangle=(v+u)/2$$: $$$\begin{gather}\Tblue{s=\frac{(u+v) t}{2}}\end{gather}$$$
- Combining the first and second results to eliminate $$v$$ or $$u$$:$$$\begin{gather}s =\frac{ut+(u+at)t}{2} \quad s =\frac{vt+(v-at)t}{2}\\\Tblue{s =ut+\frac{1}{2}at^2 \quad s = vt-\frac{1}{2}at^{2}}\end{gather}$$$
- Combining the first and second results to eliminate $$t$$:$$$\begin{gather}s=\frac{1}{2}(u+v)\frac{(v-u)}{a}\Rightarrow s=\frac{v^{2}-u^{2}}{2a}\\\Tblue{v^2=u^2+2as}\end{gather}$$$

Non-uniform acceleration is acceleration that **changes** with time.

Many real situations involve non-uniform acceleration. This can be because of **air resistance**.

Air resistance is the force on a moving object by the air. Air resistance always works to slow the object down. The **faster** an object is moving, the **greater** the air resistance.

An object falling through the air has a non-uniform acceleration. The acceleration due to gravity stays the same. The deceleration due to air resistance **increases** as the object speeds up.

The overall acceleration of the object **decreases** with time.

We often simplify real situations by **ignoring** the effects of air resistance.

When a pencil falls from the table to the ground, air resistance is normally ignored.