Speed is the distance covered in unit time. Velocity is the displacement in unit time, which is another way of saying the rate of displacement. Both are measured in metres per second. Speed is a scalar and velocity is a vector, and the reason for that difference is sitting in the topic you have just finished.
One division, two different top numbers
This is the easiest topic in the unit, provided you notice one thing before you start. Speed and velocity are not two new ideas. They are the two ideas you already have, each divided by time.
| Speed = distance ÷ time, written as \( v = \dfrac{d}{t} \) |
| Velocity = displacement ÷ time, written as \( v = \dfrac{S}{t} \) |
The bottom of both fractions is the same. Only the top changes. So every single thing that was true about distance and displacement is inherited straight upwards. Distance never shrinks, so speed is never negative. Displacement has a direction, so velocity has a direction. Distance is at least as big as displacement, so speed is at least as big as velocity. You do not have to learn any of that again. You already know it.
Dolma goes to her uncle's house
Dolma takes road A from her house to her uncle's house and it takes her 30 minutes. To find her speed we divide the total distance she walked by the time it took.
| Distance (d) = 1800 m |
| Time (t) = 30 minutes = 30 × 60 = 1800 s |
| Speed = \( \dfrac{d}{t} = \dfrac{1800}{1800} = 1 \) m/s |
Now the same journey again, measured the other way. Going by road B, the shortest distance from her house to her uncle's house, which is the displacement, is only 1200 m.
| Displacement (S) = 1200 m |
| Time (t) = 1800 s |
| Velocity = \( \dfrac{S}{t} = \dfrac{1200}{1800} = 0.67 \) m/s towards the east |
Notice two things about that pair of answers. The speed came out larger than the velocity, and it always will, for the same reason distance is always at least as large as displacement. And the velocity answer is not finished until the direction is written down. An examiner who sees 0.67 m/s with no direction is looking at an incomplete answer, because velocity is a vector.
Speed and velocity share the same unit. Both are metres per second. That is exactly why they are so easy to confuse, and it means the unit can never tell you which one you are holding. If a question hands you a number in m/s and nothing else, you cannot know whether it is a speed or a velocity by looking at it. You have to look at what was divided. If the top of the fraction was the whole path covered, it is a speed. If the top was the straight gap from start to finish, it is a velocity. Read the question, never the unit.
The four differences, written out
| Basis of difference | Speed | Velocity |
|---|---|---|
| Definition | The distance covered in unit time | The displacement in unit time, that is, the rate of displacement |
| Direction | No direction is needed to state it fully | A direction must be stated, or the answer is incomplete |
| Scalar or vector | Scalar | Vector |
| Relation to distance or displacement | Calculated from distance, so it is never negative and never zero while the object is moving | Calculated from displacement, so it can be zero or negative even while the object is moving |
A racing car can have zero velocity
Here is the consequence that makes the difference impossible to forget, and it is the walk round your house from the last topic, moved up one level.
Picture a car going flat out round a circular track. It completes one lap of 4000 m in 100 seconds. Its speed is 4000 divided by 100, which is 40 m/s, and anybody watching would agree it was moving very fast indeed. Its velocity over that lap is zero, because it finished the lap at exactly the point it started, so the displacement was nothing at all and nothing divided by 100 is still nothing.
Both numbers are correct and they are not in conflict, because they never measured the same thing. The speed describes how hard the engine worked. The velocity describes how much progress was made from where it began. Over a whole race of many laps, the car covers a huge distance at high speed and finishes with a velocity of almost nothing, because the finishing line is next to the starting line.
Everything in this topic is average speed and average velocity, worked out over a whole journey. The needle on a bus or a motorcycle is doing something slightly different: it shows how fast the vehicle is going at this exact instant, which is called the instantaneous speed. The two are only the same number if the vehicle travelled at a perfectly steady rate the whole way, which almost never happens on a real road. So when somebody says the bus averaged thirty kilometres an hour to Pokhara, it does not mean the needle ever sat at thirty. It means that the total distance divided by the total time came to thirty, after all the stopping, crawling and overtaking had been included.
Worked example 1: Rahima walks from P to R
Rahima covers the path from P to R by way of Q. It takes her 1 minute to get from P to Q and 2.5 minutes from Q to R. Find her speed. If it also took her the same time to go straight from P to R, what would her velocity be?
| Step | Working |
|---|---|
| Distance | d = 32 m + 78 m = 110 m, because she walked both sides |
| Displacement | S = 100 m, the straight line from P to R |
| Time | t = 1 min + 2.5 min = 3.5 min = 3.5 × 60 = 210 s |
| Speed | \( v = \dfrac{d}{t} = \dfrac{110}{210} = 0.52 \) m/s |
| Velocity | \( v = \dfrac{S}{t} = \dfrac{100}{210} = 0.48 \) m/s, from P towards R |
So Rahima's speed is 0.52 m/s and her velocity is 0.48 m/s from P towards R. Check the pattern one more time: the speed is the larger of the two, and only the velocity carries a direction.
Worked example 2: rearranging the formula
A car is travelling at a speed of 30 m/s. How long will it take to cover a distance of 80 km? This is the same formula, used backwards.
| Distance d = 80 km = 80 × 1000 = 80000 m |
| Speed v = 30 m/s, and time t = ? |
| From \( v = \dfrac{d}{t} \) we get \( t = \dfrac{d}{v} = \dfrac{80000}{30} = 2666.67 \) s |
| 2666.67 s ÷ 60 = 44.45 minutes |
First, convert before you divide, never after. Kilometres must become metres by multiplying by 1000, and minutes must become seconds by multiplying by 60, and both have to be done before the number goes into the formula. A student who puts 80 and 30 into the division gets 2.67, which is not a time in any unit at all. Second, look at your answer and ask whether it is sensible. Forty four minutes to cover eighty kilometres is believable for a car. If you had got forty four seconds, or forty four hours, something has gone wrong with a conversion and it is worth finding it before you move on.
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