Rate of Change

Speed is how quickly something moves: the distance travelled in each unit of time. The three quantities are linked by: $\text{speed} = \frac{\text{distance}}{\text{time}}.$

Rearranging gives the two partners you will use just as often: $\text{distance} = \text{speed} \times \text{time}, \qquad \text{time} = \frac{\text{distance}}{\text{speed}}.$

The famous triangle trick can help: cover the quantity you want, and the other two tell you what to do.

The other half of the skill is units:

• $\text{km}$ and $\text{h}$ give a speed in $\text{km/h}$.
• $\text{m}$ and $\text{s}$ give a speed in $\text{m/s}$.

Convert distances and times to match before you divide. To swap between units, $1\,\text{km/h} = \tfrac{1000}{3600}\,\text{m/s}$, so $36\,\text{km/h} = 10\,\text{m/s}$.

A distance-time graph is the simplest motion graph. Time goes on the x-axis (horizontal) and distance from a starting point on the y-axis (vertical). Every section of the graph tells you a story about how someone (or something) is moving.

• Sloping up: moving away from the start at a steady speed.
• Sloping down: moving back towards the start.
• Flat (horizontal): standing still — time passes, distance does not change.
• Steeper line: faster speed; gentler line: slower speed.

Crucially, on a distance-time graph the gradient of any straight section equals the speed at that moment: $\text{speed} = \frac{\text{change in distance}}{\text{change in time}}.$

So a sketch lets you compare speeds at a glance: the steeper the line, the faster the motion. A flat line has gradient zero, i.e. zero speed.

A speed-time graph swaps the y-axis to speed and keeps time on the x-axis. The story is now about whether the speed itself is going up, down or staying the same.

• Flat horizontal line: constant speed (not stopped — moving at a steady pace).
• Sloping up: speeding up, i.e. accelerating.
• Sloping down: slowing down, i.e. decelerating.
• Reaching the x-axis (speed = 0): object is now stopped.

Watch out — the shape of a flat line on the two graphs means opposite things:

• Flat on a distance-time graph = standing still.
• Flat on a speed-time graph = moving at a steady speed.

You can also read total distance off a speed-time graph by finding the area under the line, but for Checkpoint the most useful skills are simply recognising the three behaviours above and the units on the axes.

The gradient of a straight line measures how much $y$ changes for every unit increase in $x$. Pick two clear points $(x_1, y_1)$ and $(x_2, y_2)$ on the line; then: $\text{gradient} = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}.$

That number is a rate of change — and its meaning depends on the graph:

• Distance vs time → gradient is speed (e.g. metres per second).
• Speed vs time → gradient is acceleration (e.g. metres per second every second).
• Cost vs items → gradient is price per item.
• Water vs minutes → gradient is flow rate (e.g. litres per minute).

A positive gradient slopes up to the right; a negative gradient slopes down to the right; a zero gradient is flat. Whenever a Checkpoint question asks what the gradient means, answer with the units — say "the speed is $2\,\text{m/s}$", not just "the gradient is $2$".

Real journeys often have stages: running, resting, walking. The average speed for the whole journey is not the average of the individual speeds. It is the total distance divided by the total time: $\text{average speed} = \frac{\text{total distance}}{\text{total time}}.$

This includes all the resting time, which is exactly why the average is usually slower than each of the moving stages.

A worked breakdown:

• Stage 1: run $400\,\text{m}$ in $100\,\text{s}$ (speed $= 4\,\text{m/s}$).
• Stage 2: rest for $50\,\text{s}$ (distance $= 0$).
• Stage 3: walk $200\,\text{m}$ in $100\,\text{s}$ (speed $= 2\,\text{m/s}$).

Total distance $= 400 + 0 + 200 = 600\,\text{m}$. Total time $= 100 + 50 + 100 = 250\,\text{s}$. Average speed $= 600 \div 250 = 2.4\,\text{m/s}$.

Notice how this is less than either of the moving speeds — the rest pulled the average down. A distance-time graph for this journey would have three sections: a steep line, a flat line, then a less-steep line. Reading the y-axis at the end gives the total distance; reading the x-axis gives the total time.

Speed and the gradient idea sit at the heart of mathematics for the rest of your school years and beyond.

• Linear equations in IGCSE use gradient as the $m$ in $y = mx + c$ — the same rise-over-run idea.
• Acceleration in physics is the gradient of a speed-time graph and is fundamental to mechanics.
• Calculus (in A Level) takes "rate of change" to a precise level using derivatives and tangents to curves.
• Real-world applications include population growth, flow rates, currency exchange rates, and reading any business graph.

Whenever you see a graph, ask yourself: what are the axes, and what does the gradient mean in their units? That habit will make every graph you ever meet much easier to read.