HSC Maths Advanced Syllabus

Motion in Various Dimensions

The direction of motion can occur in various dimensions. Motion in one dimension refers to motion that is travelling in one line. This can be seen through the variable $x$ when written as a function of time (denoted as $t$). The variable $x$ refers to the displacement of an object on a number line.

It is important to know that time can’t be a negative. When a question specifies the use of negative time, it generally means time is travelling backwards. As such, it is highly unlikely to find a question dealing with negative time.

What is Displacement?

Displacement represents the change in position of an object. It is not the same as distance; displacement measures the shortest path between the start and end points, considering direction. Displacement is noted as the letter $x$, while the change in displacement is represented by:

$$\Delta x=x_{final}-x_{intial}$$

where $x_{final}$ is the final position of an object, and $x_{intial}$ is the object’s initial position.

Average Velocity

Average velocity is a measure of the speed of the movement of an object in a specific direction. It is a vector quantity, which means it has both magnitude and direction. For example, if a car travels north for 100 meters in 5 seconds, its average velocity is 20 meters per second north. Thus, it is defined by the formula:

$$V_{average}=\frac{x_2-x_1}{t_2-t_1}=\frac{\Delta x}{\Delta t}$$

Where, $x_2-x_1$ is the change in an objects displacement, and $t_2-t_1$ is the change in time, or time it took the object to travel from $x_1$ to $x_2$.

Average Speed

Average speed is a scalar quantity that measures how fast an object is moving. Unlike average velocity, it does not account for the direction of travel. For instance, if a person walks around a block covering a total distance of 1 km in 10 minutes, their average speed is 6 km/h , regardless of their starting or ending point. Thus, its formula is:

$$S_{average}=\frac{total\:distance\:travelled}{time\:taken(t)}$$

It is also important to note, that because direction is not involved when calculating average speed, it can never be a negative value.

Similar to problems involving average velocity, we must ensure that the units used to calculate equations are consistent. For example, it is easiest to use $km$ and $hours$ together, while $meters$ and $seconds$ are generally used together.