Speed–Distance–Time (Rearrange)

\( s=\tfrac{d}{t},\quad d=st,\quad t=\tfrac{d}{s} \)
Measures GCSE
Question 2 of 20
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A cyclist rides 24 km at 12 km/h. Find the time.

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
\( t = d ÷ s \)

Explanation

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Statement

The relationship between speed, distance, and time is one of the most fundamental formulas in mathematics and physics. Speed is distance divided by time, distance is speed multiplied by time, and time is distance divided by speed.

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

Why it’s true

  • Speed measures how far something travels in a given time interval.
  • If you know speed and time, multiplying them gives the total distance covered.
  • If you know distance and time, dividing gives the average speed.
  • If you know distance and speed, dividing gives the time taken.

Recipe (how to use it)

  1. Identify which two quantities are given and which one is missing.
  2. Choose the correct rearranged version of the formula.
  3. Substitute the known values, making sure units are consistent (metres with seconds, or kilometres with hours).
  4. Calculate and state the result with correct units.

Spotting it

These problems usually involve a journey or a moving object, with two values given (speed, distance, or time) and one missing. Keywords like “how long”, “how far”, or “at what speed” signal which form of the formula is needed.

Common pairings

  • Conversion between km/h and m/s before applying the formula.
  • Multi-step word problems where one part requires distance, the next requires time.
  • Graph questions involving distance–time or speed–time graphs.

Mini examples

  1. Given: Distance = 120 km, Time = 2 h. Find speed: \(120 ÷ 2 = 60\) km/h.
  2. Given: Speed = 5 m/s, Time = 40 s. Find distance: \(5 × 40 = 200\) m.

Pitfalls

  • Forgetting to convert units (e.g. hours into seconds).
  • Mixing up whether to multiply or divide.
  • Leaving answers without units.
  • Using inconsistent units (e.g. km with seconds).

Exam strategy

  • Write down the correct formula triangle (speed–distance–time).
  • Check what is asked: “how far”, “how fast”, or “how long”.
  • Convert units before calculating if necessary.
  • Always label the final answer with the correct unit.

Summary

The speed–distance–time triangle is a versatile tool for GCSE maths and physics. By rearranging the base formula \(s = \frac{d}{t}\), you can solve for whichever quantity is missing. Consistent units and careful rearrangement are the keys to success.

Worked examples

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  1. A car travels 150 km in 3 hours. Find its average speed.
    1. \( s = d / t \)
    2. \( s = 150 / 3 \)
    3. \( s = 50 \)
    Answer: 50 km/h
  2. A runner covers 400 m in 80 seconds. Find the speed.
    1. \( s = d / t \)
    2. \( s = 400 / 80 \)
    3. \( s = 5 \)
    Answer: 5 m/s
  3. A cyclist moves at 20 km/h for 2 hours. Find the distance.
    1. \( d = s × t \)
    2. \( d = 20 × 2 \)
    3. \( d = 40 \)
    Answer: 40 km
  4. A person walks 5 km at 4 km/h. Find the time taken.
    1. \( t = d / s \)
    2. \( t = 5 / 4 \)
    3. \( t = 1.25 \)
    Answer: 1.25 h (1 h 15 min)
  5. A bus travels 180 km in 4 hours. Find its average speed.
    1. \( s = d / t \)
    2. \( s = 180 / 4 \)
    3. \( s = 45 \)
    Answer: 45 km/h
  6. A motorbike travels 300 km at 75 km/h. Find the time taken.
    1. \( t = d / s \)
    2. \( t = 300 / 75 \)
    3. \( t = 4 \)
    Answer: 4 h
  7. A train travels at 25 m/s for 3 minutes. Find the distance.
    1. \( t = 3 × 60 = 180 s \)
    2. \( d = s × t \)
    3. \( d = 25 × 180 \)
    4. \( d = 4500 \)
    Answer: 4500 m
  8. A car moves at 90 km/h for 40 minutes. Find the distance.
    1. \( t = 40 min = 2/3 h \)
    2. \( d = 90 × 2/3 \)
    3. \( d = 60 \)
    Answer: 60 km
  9. A runner covers 2 km in 8 minutes. Find the average speed in m/s.
    1. \( d = 2000 m, t = 8 × 60 = 480 s \)
    2. \( s = 2000 / 480 \)
    3. ≈ 4.17
    Answer: ≈ 4.17 m/s
  10. A jet travels 1200 km in 2 hours. Find the average speed in m/s.
    1. \( d = 1200 km = 1,200,000 m \)
    2. \( t = 2 × 3600 = 7200 s \)
    3. \( s = 1,200,000 / 7200 ≈ 166.7 \)
    Answer: ≈ 166.7 m/s