Distance, Speed & Time

Core Formulas and Rearranging

Speed measures how quickly an object covers distance. All calculations connect through the formula $d = st$, where $d$ is distance, $s$ is speed, and $t$ is time.

At Higher Level, change the subject of the formula using algebra rather than relying on formula triangles:

• To find distance: $d = st$
• To find speed: divide both sides by $t$ to get $s = \frac{d}{t}$
• To find time: divide both sides by $s$ to get $t = \frac{d}{s}$

Proportionality connects these variables directly:

• At a constant speed, distance is directly proportional to time: double the time means double the distance.
• For a fixed distance, speed and time are inversely proportional: double the speed means half the time.

Units, Conversions, and Accuracy

Units must match before substituting values into any formula. Standard pairs are kilometres with hours ($\text{km/h}$) and metres with seconds ($\text{m/s}$).

To convert units:

• Hours and minutes to hours: divide the minutes by $60$. For example, $1\text{ h } 40\text{ min} = 1 + \frac{40}{60} = \frac{5}{3}\text{ hours}$. Keep awkward decimals as fractions so you do not lose accuracy.
• $\text{km/h}$ to $\text{m/s}$: multiply by $1000$ and divide by $3600$ (or divide by $3.6$). For example, $90\text{ km/h} = \frac{90 \times 1000}{3600} = 25\text{ m/s}$.
• $\text{m/s}$ to $\text{km/h}$: multiply by $3600$ and divide by $1000$ (or multiply by $3.6$). For example, $20\text{ m/s} = 20 \times 3.6 = 72\text{ km/h}$.
• Speed per second to speed per minute: multiply by $60$.

Keep full calculator accuracy during working, write the required unit, and round only at the end to the requested accuracy or scientific notation $a \times 10^n$.

Average Speed for Multi-Stage Journeys

For any multi-stage journey, average speed is calculated using overall totals:

Averaging stage speeds generally gives the wrong answer. If you travel two stages of equal distance at different speeds, you spend longer travelling at the slower speed, so the slow part counts for more of the total journey time.

For example, driving $60\text{ km}$ at $60\text{ km/h}$ takes $1\text{ hour}$, and driving another $60\text{ km}$ at $30\text{ km/h}$ takes $2\text{ hours}$. The total distance is $120\text{ km}$ and the total time is $3\text{ hours}$. The average speed is $\frac{120}{3} = 40\text{ km/h}$, not the arithmetic mean of $45\text{ km/h}$.

Drawing and Interpreting Distance–Time Graphs

Always check axis labels and units first. With time horizontally and total distance travelled vertically, slope gives speed: change in distance ÷ change in time.

Key graph rules:

• A steeper line means a faster speed.
• A flat, horizontal segment means the object is stationary (speed is zero).
• Distance travelled can never decrease. On a distance-from-home graph, a downward segment shows returning home. For travel along a straight route towards or away from home, speed is the positive size of the slope.
• On a distance-from-home graph, the intercept shows starting distance, the highest point shows greatest distance, and zero means home.

To draw a graph:

• Label axes with quantities and units, using an even scale.
• Plot the coordinates for each time–distance pair.
• Join points with straight segments for periods of constant speed.

Key terms

Speed

The distance travelled per unit of time, calculated as $s = \frac{d}{t}$ and measured in units such as $\text{km/h}$ or $\text{m/s}$.

Average Speed

The total distance travelled divided by the total elapsed time for an entire journey, including any stops.

Constant Speed

Motion where the rate of travel does not change, shown by a straight line segment on a distance–time graph.

Stationary

The state of being at rest, represented on a distance–time graph by a horizontal line segment with a slope of zero.

Slope (rate of change)

Vertical change ÷ horizontal change; with time horizontally and total distance travelled vertically, slope gives speed.