Distance–Speed–Time

Statement

The relationship between distance (\(d\)), speed (\(s\)), and time (\(t\)) is fundamental in motion problems. The formulas are:

\[ s = \frac{d}{t}, \quad d = st, \quad t = \frac{d}{s} \]

This shows that speed is distance divided by time, distance is speed multiplied by time, and time is distance divided by speed.

Why it’s true

• Speed is defined as how much distance is covered per unit of time.
• Rearranging the formula gives distance and time in terms of the other two quantities.
• This set of formulas is consistent with units: for example, if speed is in km/h and time in hours, distance is in km.

Recipe (how to use it)

1. Identify which two of distance, speed, and time are given.
2. Choose the appropriate formula to calculate the missing one.
3. Substitute values carefully with consistent units (convert minutes to hours, metres to kilometres, etc.).
4. Perform the calculation step by step.

Spotting it

This formula applies whenever you are asked to calculate how far, how fast, or how long, given two of the three values.

Common pairings

• Journey problems in word problems.
• Conversions of speed (e.g., km/h to m/s).
• Physics problems involving velocity and uniform motion.

Mini examples

1. Given: A car travels 120 km in 2 hours. Find speed: \(s = \frac{120}{2} = 60 \text{ km/h}\).
2. Given: A train moves at 80 km/h for 3 hours. Find distance: \(d = 80 \times 3 = 240 \text{ km}\).
3. Given: A cyclist covers 50 km at 25 km/h. Find time: \(t = \frac{50}{25} = 2 \text{ hours}\).

Pitfalls

• Forgetting to use consistent units (e.g., mixing minutes with hours).
• Using average speed incorrectly when speeds change.
• Confusing rearrangements (remember the triangle trick: distance on top, speed and time below).

Exam strategy

• Always write the formula before substituting values.
• Convert all units at the start to avoid mistakes later.
• Check if the answer makes sense (e.g., time shouldn’t be negative, speed shouldn’t be unreasonable).

Summary

The distance–speed–time formula is one of the most commonly used equations in mathematics and physics. By remembering the relationship \(s = \tfrac{d}{t}\), you can quickly rearrange to find distance or time when needed, as long as units are handled consistently.