Speed, Time and Distance: Formula, Conversions and Train

Speed is how fast something moves, defined as the distance covered in one unit of time. The formula is Speed = Distance ÷ Time, so if a bus covers 120 kilometres in 2 hours, its speed is 60 kilometres per hour. Everything else in this chapter — relative speed, train problems, boats and streams — is built on this one relationship.

Speed matters because nearly every journey, race and schedule depends on it. Get the formula and its unit conversions right, and these problems become the easiest marks in any exam.

Key Concepts

• Speed (v): distance per unit time. Common units: kilometres per hour (km/h) and metres per second (m/s).
• Distance (d): total length travelled, in km or m.
• Time (t): duration of travel, in hours or seconds.
• Average speed: total distance ÷ total time (not the average of two speeds).
• Relative speed: for two bodies moving towards each other, add their speeds; moving in the same direction, subtract the smaller from the larger.
• Conversion: 1 km/h = 5/18 m/s; 1 m/s = 18/5 km/h.

Formulas

• v = d / t, so d = v × t and t = d / v. Units must match: if v is in km/h, d must be in km and t in hours.
• km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5.
• Train crossing a pole or person: time = length of train ÷ speed of train.
• Train crossing a platform or bridge: time = (length of train + length of platform) ÷ speed of train.
• Average speed = total distance ÷ total time. Assumes the whole journey is one trip; it is not the simple mean of speeds.

Worked Example

A train 180 metres long runs at 72 km/h. How long does it take to cross an electric pole?

1. Convert speed to m/s: 72 × 5/18 = (72 ÷ 18) × 5 = 4 × 5 = 20 m/s.
2. A pole has no length, so distance = train length = 180 m.
3. Time = distance ÷ speed = 180 ÷ 20 = 9 seconds.

The train crosses the pole in 9 seconds. Units check: metres ÷ (metres/second) = seconds, so the setup is consistent.

Common Misconceptions

• "Average speed is the average of the two speeds." A car going 40 km/h one way and 60 km/h back over the same distance does not average 50 km/h. Total distance 2d, total time d/40 + d/60 = 5d/120, so average speed = 2d ÷ (5d/120) = 48 km/h. More time is spent at the slower speed, so it pulls the average down.
• "Multiply by 5/18 to convert m/s to km/h." Backwards. Going km/h to m/s, multiply by 5/18; going m/s to km/h, multiply by 18/5. Test with a memory anchor: 36 km/h = 10 m/s.
• "Crossing a platform uses only the train's length." No — the train has fully crossed only when its tail leaves the platform, so add both lengths.

Real-World Applications

• Railways and airlines build timetables and announce delays from these formulas.
• Navigation apps estimate your arrival time (ETA) using current speed and remaining distance.
• Athletes and coaches plan race pacing from target speeds.

Speed, time and distance is one formula worn three ways: v = d/t, d = v × t, t = d/v. Keep the units matched, convert with 5/18 and 18/5 in the right direction, and remember that crossing problems add the lengths of everything the train must pass.