Average Velocity and Speed (HSC SSCE Mathematics Advanced): Revision Notes

Average Velocity and Speed

Motion in one dimension

When studying the motion of an object along a straight line, we describe its position using displacement. Displacement is a single value, denoted by $x$, that tells us where the object is located at any given time $t$.

The entire motion can be described mathematically by expressing displacement as a function of time. This allows us to track the object's position throughout its journey.

Example: A ball thrown vertically upward

Consider a ball that is thrown straight up from ground level and lands back on the ground 8 seconds later. We can model this motion using the equation:

$x = 5t(8 - t)$

where $x$ represents the height in metres above the ground, and $t$ represents the time in seconds after the ball is thrown.

Key features of this motion:

• The ball starts at ground level when $t = 0$ (point O)
• It reaches maximum height of 80 metres at $t = 4$ seconds (point B, the vertex)
• It returns to ground level at $t = 8$ seconds (point D)
• The domain is $0 \leq t \leq 8$ and the range is $0 \leq x \leq 80$

Average velocity

Average velocity measures how quickly displacement changes over a time interval. It can be positive (moving in the positive direction), negative (moving in the negative direction), or zero (no net change in position).

The formula for average velocity is:

$\text{Average velocity} = \frac{\text{change in displacement}}{\text{change in time}} = \frac{x_2 - x_1}{t_2 - t_1}$

where $x_1$ is the displacement at time $t_1$ and $x_2$ is the displacement at time $t_2$.

Graphical interpretation

On a displacement-time graph, the average velocity equals the gradient of the chord connecting the two points representing the start and end of the time interval.

Example from the ball's motion:

During the ball's ascent (from $t = 0$ to $t = 4$):

• Change in displacement = $80 - 0 = 80$ metres
• Change in time = $4 - 0 = 4$ seconds
• Average velocity = $\frac{80}{4} = 20$ metres per second

This equals the gradient of chord OB on the graph.

During the ball's descent (from $t = 4$ to $t = 8$):

• Change in displacement = $0 - 80 = -80$ metres
• Change in time = $8 - 4 = 4$ seconds
• Average velocity = $\frac{-80}{4} = -20$ metres per second

The negative value indicates downward motion. This equals the gradient of chord BD.

Distance travelled vs displacement

It's essential to understand the difference between change in displacement and distance travelled.

Example: In our ball example, during the time from $t = 4$ to $t = 8$:

• Change in displacement = $0 - 80 = -80$ metres
• Distance travelled = 80 metres (the ball moves down 80 metres)

For the complete 8-second flight:

• Change in displacement = $0 - 0 = 0$ metres (ball returns to starting position)
• Total distance travelled = $80 + 80 = 160$ metres (up 80m, then down 80m)

Average speed

Average speed measures how much distance is covered per unit of time. Unlike average velocity, average speed can never be negative.

$\text{Average speed} = \frac{\text{distance travelled}}{\text{time taken}}$

Key difference from average velocity

For the complete flight of our ball (8 seconds):

Average velocity = $\frac{0 - 0}{8 - 0} = 0$ m/s

Average speed = $\frac{160}{8} = 20$ m/s