Use the equations of motion to analyse uniform and uniformly accelerated straight line motion.

What this dot point is asking

This dot point is the foundation of mechanics. Before you can analyse forces, momentum or projectiles you must be able to describe how a single object moves along a line and predict where it will be and how fast it will be travelling.

Vector quantities of motion

Position, displacement, velocity and acceleration are all vectors, so direction matters. In one dimension you handle direction with a sign: choose one direction as positive and the opposite as negative, then keep that convention for the whole problem.

• Displacement $s$ is the change in position, a straight line from start to finish, not the distance travelled along the path.
• Velocity $v$ is the rate of change of displacement, so average velocity is $\dfrac{\Delta s}{\Delta t}$.
• Acceleration $a$ is the rate of change of velocity, $\dfrac{\Delta v}{\Delta t}$.

Speed and distance are the scalar partners of velocity and displacement. A car that drives a loop and returns has travelled a distance but has zero displacement, so its average velocity is zero even though its average speed is not.

The equations of motion

When acceleration is constant, five quantities are linked by four equations. The symbols are initial velocity $u$, final velocity $v$, acceleration $a$, displacement $s$ and time $t$:

Each equation leaves out one variable, so the trick is to list what you know, identify the one quantity you do not need, and pick the equation that omits it. This avoids unnecessary algebra.

Free fall and vertical motion

Near Earth's surface, objects in free fall accelerate downward at $g = 9.8\ \text{m s}^{-2}$ regardless of mass, once air resistance is ignored. Vertical motion is just the equations of motion with $a = g$. Choosing up as positive means $a = -9.8\ \text{m s}^{-2}$, and at the top of a thrown object's flight the velocity is momentarily zero while the acceleration is still $g$ downward.

Reading motion graphs

Graphs carry the same information as the equations. On a displacement-time graph the gradient is velocity. On a velocity-time graph the gradient is acceleration and the area under the line is displacement. A straight sloped line on a velocity-time graph means constant acceleration; a curve means changing acceleration. Being able to switch between graphs and equations is a frequently tested skill.

In the exam, write down your sign convention, list the five symbols with their known values, then choose the single equation that contains your three knowns and one unknown. Check that your answer has a sensible sign and units before moving on.