Speed, distance and time: the formula and its traps
The speed, distance and time formula in all three forms, how to convert between km/h, m/s, mph and knots, why average speed is not the average of two speeds, and how exam problems on trains work.
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In short
- Speed = distance ÷ time, distance = speed × time, and time = distance ÷ speed.
- To turn km/h into m/s, multiply by 5/18. To go back, multiply by 18/5. So 72 km/h is 20 m/s.
- Average speed is total distance ÷ total time. Going 60 km/h out and 40 km/h back averages 48, not 50.
- 2.5 hours is 2 hours 30 minutes, but 2 hours 30 minutes on a clock is not 2.3 hours.
- For a train passing something, the distance is the train's own length plus the length of the object.
The relationship between speed, distance and time is one of the first formulas anyone meets in physics, and one of the few used every day. It tells you when to leave for a train, whether a delivery estimate is honest and how fast you ran this morning. In exams it appears as train, boat and relative speed problems. The formula itself fits on one line. The mistakes come from units, from time written in hours and minutes, and from average speed. This page covers all of them.
The formula
speed = distance ÷ time
Rearranged, that gives the other two:
distance = speed × time
time = distance ÷ speed
If you cover one word in the triangle of D at the top with S and T beneath, the other two show you what to do. Cover S and you see D over T. Cover D and you see S beside T.
For example, 150 km in 2.5 hours is a speed of 60 km/h. At 75 km/h, 450 km takes 6 hours. At 80 km/h for 3 hours, you cover 240 km.
Keep the units consistent
Before doing anything else, make the units agree. If speed is in km/h, the distance must be in km and the time in hours. The calculator converts for you, but by hand this is where most errors occur.
| From | To | Multiply by |
|---|---|---|
| km/h | m/s | 5/18, about 0.2778 |
| m/s | km/h | 18/5, which is 3.6 |
| mph | km/h | 1.609 |
| knots | km/h | 1.852 |
| km/h | mph | 0.6214 |
The 5/18 comes from the units themselves: 1 km is 1,000 m and 1 hour is 3,600 s, and 1,000 ÷ 3,600 = 5/18. So 36 km/h is 10 m/s, 54 km/h is 15 m/s, and 72 km/h is 20 m/s. These are worth remembering for exams.
A knot is one nautical mile per hour, and it is used for ships and aircraft. A nautical mile is 1.852 km.
Time in hours and minutes
Clocks do not use decimals, and this catches people out. Thirty minutes is half an hour, so 2 hours 30 minutes is 2.5 hours, not 2.3.
| Minutes | Decimal hours |
|---|---|
| 6 | 0.1 |
| 15 | 0.25 |
| 20 | 0.333 |
| 30 | 0.5 |
| 40 | 0.667 |
| 45 | 0.75 |
To convert, divide the minutes by 60. The other way, multiply the decimal part by 60. So 1.2 hours is 1 hour 12 minutes, and 3.75 hours is 3 hours 45 minutes. The calculator shows times in both forms.
The average speed trap
You drive 120 km at 60 km/h and return at 40 km/h. What is your average speed? It is natural to say 50. The real answer is 48.
The outward trip takes 2 hours and the return takes 3, so you have covered 240 km in 5 hours, and 240 ÷ 5 = 48 km/h. The slower leg takes longer, so it counts for more.
average speed = total distance ÷ total time
When the two distances are equal, there is a shortcut:
average speed = 2 × v₁ × v₂ ÷ (v₁ + v₂)
Here that is 2 × 60 × 40 ÷ 100 = 48. Note that the distance does not matter, so the answer would be the same for 10 km each way. If you spend equal time at each speed, however, the simple average is correct.
This explains a surprise on long journeys. Traffic that halves your speed for a stretch costs more time than driving faster elsewhere can recover. If a trip is 100 km and the first 50 km take a full hour, reaching a 100 km/h average would need the second half to take no time at all.
Relative speed
When two things move, what matters is how fast the gap changes.
Moving towards each other. Add the speeds. Two trains at 60 and 40 km/h, 300 km apart, close at 100 km/h, so they meet after 3 hours.
Moving the same way. Subtract. A car at 80 km/h behind a truck at 60 km/h gains at 20 km/h, so a gap of 5 km closes in 15 minutes.
Train problems
These are favourites in aptitude tests. The key point is that a train has length.
Passing a pole or a person. The train must travel its own length. A 200 m train at 72 km/h, which is 20 m/s, passes a pole in 200 ÷ 20 = 10 seconds.
Passing a platform. The train must travel its own length plus the platform's. With a 300 m platform, that is 500 m, which takes 25 seconds.
Boats and streams. Downstream, the current adds to the boat's speed. Upstream, it subtracts. A boat at 10 km/h in still water, on a 2 km/h stream, goes at 12 km/h and 8 km/h.
Everyday uses
Planning a trip. Divide the distance by a realistic average speed, not the speed limit. Indian highway trips often average much less than the limit, once tolls and town stretches are counted. Fuel for the same trip comes from the fuel cost calculator.
Running and walking. Runners prefer pace, which is time per kilometre. Five kilometres in 30 minutes is 10 km/h, or 6 minutes per km. To convert, divide 60 by the speed in km/h.
Road safety. At 72 km/h, a car travels 20 m every second. Looking at a phone for two seconds means 40 m driven unseen, and that is before any braking distance. Speed matters more than people think, because the energy that a car carries grows with the square of its speed.
Sound and storms. Sound travels about 340 m per second in air. Count the seconds between lightning and thunder and multiply by 340 to estimate the distance to a storm. Three seconds is about a kilometre.
Worked examples from daily life
Catching a train. Your station is 18 km away, and traffic averages 24 km/h. The ride takes 18 ÷ 24 = 0.75 hours, which is 45 minutes. Add time for parking and the platform, then count back from the departure time.
Checking a delivery estimate. A truck must cover 1,400 km, and a driver averages 50 km/h for 10 hours a day. That is 500 km a day, so the trip takes 2.8 days, which means three days on the road.
Reading a flight's speed. A plane covers 1,150 km from Delhi towards Mumbai in about 1 hour 30 minutes in the air. That is 1,150 ÷ 1.5, or about 767 km/h. The time on your ticket is longer, since it includes taxiing and waiting.
Finding a missing time. A cyclist rides 12 km at 15 km/h. The time is 12 ÷ 15 = 0.8 hours, which is 48 minutes.
How to approach an exam problem
Write down what you know, with its unit, before you touch the formula. Change everything into one set of units. Decide which of the three is missing, then pick the matching form of the formula. Finally, ask whether the answer is sensible. A person walking does not go 40 km/h, and a train does not take 10 seconds to cover 100 km.
What this calculator assumes
- The speed is constant over the journey, or is the average speed for it.
- Stops are counted as part of the time only if you include them.
- One mile is 1.609344 km, and one nautical mile is 1.852 km.
- The answer is shown in your chosen unit, with the matching values in other units.
- It treats movement along a line, and does not allow for acceleration.