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# Uniform motion and uniform acceleration

## 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.

Parachutists free falling is an example of uniform acceleration if air resistance is ignored.

## Formulae for uniform acceleration (SUVAT)

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.

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

## Air resistance and non-uniform acceleration

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.

Cyclists race in line behind each other to reduce 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.