Average Speed vs Average Velocity vs Instantaneous Velocity

Two taxi rides: which is faster? (preview)

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But in high school, we split 'fast' into two meanings. Saying the first was faster isn't wrong. The first one gets there sooner. Second faster? Also right: it covers ground faster,

In this lesson: four speed and velocity quantities

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In this video, we'll go through all four speed-related quantities in high school physics in one go. If you can't tell velocity or speed, or you're given some descriptions and can't tell if each is average or instantaneous? Is instantaneous speed the magnitude of instantaneous velocity? Is average speed the magnitude of average velocity? If these questions trouble you, listen to this lesson carefully. By the end, you'll have all four down:

What "fast" meant in middle school

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what each one means, differences and common mistakes. I'll explain it all. We'll also pin down the definition of the simplest, most basic kind of motion in high school: constant-velocity motion. On speed, many new high schoolers wonder: isn't speed just about fast versus slow? Easy in middle school. Why four quantities in high school? In middle school, speed meant fast versus slow. To be exact, it described how fast you cover ground. Take running. Bolt is faster than you or me, and a plane beats a train. So in middle school, 'faster' just meant whose legs move faster. Obvious and simple. A very simple, pure, beautiful world. But in high school, this world gets complicated. Even simple 'who is faster?' is trickier than before. For example, a high-school-level question:

Two taxi rides: arriving vs. running

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I took a taxi twice from home to Peking University: start and end. The same both times. First time, the driver took Yuanda Road and Wanquanhe Road, then the North 4th Ring Road. The trip was 7.5 km and took 14 minutes. Second time: straight onto 4th Ring Road. It took 15 minutes, 10 km. Of my two trips to Peking University, which one was faster? At a glance, you'd say the first, it took less time. I left at 3 p.m. both times. First trip: arrived 3:14. Second: 3:15. So the first was a minute faster. The first wins. But if you do a little math now, Divide 7.5 km by 14 minutes to see how far I go per minute on average. The first is about 0.54 km per minute. On the first trip, 7.5 km in 14 minutes, that's 0.54 km per minute. Second trip, same method: it's 0.66 km per minute. On the second trip, I averaged 0.66 km per minute.

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Looking at that, the second trip seems to cover more distance each minute. The car moved faster, so the second trip is faster. So which is faster? Not obvious now. By the middle school definition, the second is definitely faster, because its wheels turn faster, so it covers more each minute: faster. But in high school, we split 'fast' into two meanings. Saying the first was faster isn't wrong. The first one gets there sooner. Second faster? Also right: it covers ground faster, Exactly middle school's idea of fast, which in high school corresponds to a quantity called average speed. What you called 'speed' in middle school is defined the same way as average speed. As for 'getting there sooner,' high school calls that average velocity. So the 'speed' you learned in middle school and high school's average velocity differ in concept, definition and meaning.

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Another difference from middle school. These are the middle two of our four quantities. One by one, average velocity first.

Average velocity

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Average velocity: how fast you get there. Formally, how fast position changes and in what direction. Say you and I both start at point A, and leave at the same time, both heading for point B. Whoever reaches B first has the greater average velocity. Whether you and I went straight, on a winding road, or around a big loop, makes no difference. So, what's the best way to define and calculate the magnitude of average velocity? It has nothing to do with your route, so the simplest choice is to take a straight line. I don't care which route you took. I just divide the displacement by time. The shorter your time, the greater your average velocity. So it's path-independent. The magnitude of average velocity equals the straight line from start to end, divided by time. The key point about average velocity: it's independent of the actual trajectory. I only use the straight line from start to end and the time taken.

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Formally: displacement divided by time. The straight line between two points is exactly the magnitude of displacement. Average velocity = displacement / time, v = x/t, in m/s. Average velocity has a direction, that of the displacement. Because v = x / t, time is a scalar, and a vector over a scalar keeps its direction. So v and x point the same way. So A-to-B average velocity points along the arrow from A to B. That's how average velocity is defined.

Examples: the dog's laps and the cat's runs

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Let's do two examples. Longtime fans know I have a dog named Goutiao, and a cat named Goumao. Goumao isn't a dog plus a cat, just a name; both begin 'Gou' (dog). Let's see them move. Goutiao ran 10 laps of the complex, each lap 200 m, in 20 minutes. Average velocity? It ran a lot, and got tired: 10 laps of 200 m, 2000 m in total. But they were ten full laps, so after ten loops it's back at the start, isn't it? After ten laps its position is unchanged. No change in position, so average velocity is 0. It measures how fast position changes. No change in position? Definitely zero. Now Goumao. It starts at this point. Cat owners know cats love shuttle runs at home. Who knows why. It first ran 2 m to the right, then 6 m to the left. The total time is 20 s. Average velocity?

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Again, average velocity is about change in position. Look: from this point to this point, how many meters did it change? It changed this much: 4 m. So I divide this 4 m by the time, which is 0.2 m/s. Also, average velocity is a vector, so state the direction: to the left. So Goumao's average velocity is 0.2 m/s to the left. From these two examples, note that average velocity ignores the process, meaning the actual trajectory. You only look at start and end. Only the result matters. This is the core of average velocity. Also: velocity is a vector. If they ask for the magnitude, just give the value. If it just asks for the velocity, give magnitude and direction. Missing the direction costs you points. Next, a multiple-choice question.

Quiz: what average velocity tells you

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It's about average velocity. Which statement is correct? A: the car's average velocity at the end of 10 s is 5 m/s. Is this correct? First: 'the end of 10 s'. When we covered intervals and instants, we said the end of 10 s is an instant, a single point in time. Think about the real world: there's no teleporting. Goku in Dragon Ball teleports: whoosh! In everyday life, nobody teleports. However fast a position changes, it still takes some time. It's never sudden. So average velocity is always about a time interval, not an instant. Why not an instant? Because, as I said, you can't teleport. So this 5 m/s is not an average velocity. Average velocity is about a time interval. This 5 m/s, which we'll learn later, is actually an instantaneous velocity. So A is wrong. B: over some time interval, the average velocity is 5 m/s, meaning in every 1 s of it, the displacement is 5 m.

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No. Average velocity records the result, not the process. Example: points A and B are 25 m apart. If I move 5 m every second: 1, 2, 3, 4, 5. 5 m in every second. So average velocity = 25 m ÷ 5 s = 5 m/s. But I could also do this: stand still for the first 4 s, then move all 25 m in the 5th second. 25 m within one second. Average velocity over the 5 s: again 25 m ÷ 5 s, also 5 m/s. So the average velocity is 5 m/s. It only tells you that in this interval your displacement was 25 m. Steady the whole way, or fast then slow, or slow then fast? It can't tell you. Got it? So B is wrong. Average velocity can't show the process, like '5 m in every second'. It only shows the result: 25 m in 5 s.

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Nothing else. C: average velocity only roughly reflects how fast an object moves. C is correct. As I said, it only shows the result, so it's only rough. D: to find average velocity, you must consider how velocity changes. Average velocity is displacement ÷ time; changes in velocity don't matter. So D is wrong. The only answer is C. That's it for average velocity. Next concept: average speed. It describes how fast you cover ground.

Average speed

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Not much to say here: it's the speed from middle school: distance / time, how fast you cover ground over an interval. A scalar: magnitude only, no direction. One thing to watch out for: many students think average speed is just average velocity without the sign, the magnitude of average velocity. That's wrong. They mean different things. Average speed is distance ÷ time; average velocity is displacement ÷ time. Distance and displacement differ! Let's compute both and compare.

Average velocity vs. average speed

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Goumao (the cat) runs 2 m to the right, then 6 m to the left, 20 s in total. Its average velocity and average speed? First, average velocity. Displacement (start to end) ÷ time. 2 m right, then 6 m left, displacement: 6 − 2 = 4 m, 4 m ÷ 20 s = 0.2 m/s, to the left. State the direction! Now the average speed. Average speed uses distance ÷ time. What's the distance? 2 m, then 6 m: add them up, 8 m ÷ 20 s = 0.4 m/s. Magnitude of average velocity: 0.2 m/s, the average speed: 0.4 m/s. They're not equal. Now the second example. Goutiao (the dog), chasing Goumao, runs 10 m right in 5 s. Its average velocity and average speed? Same as before: average velocity first. It moves in a straight line, 10 m, so 10 m ÷ 5 s = 2 m/s.

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Now the average speed. For average speed, just use distance ÷ time: again 10 m ÷ 5 s = 2 m/s. So in the second example, the magnitude of average velocity equals average speed. I left out its direction: to the right. Never leave out the direction! Here they're equal. In fact, here's a rule: in one-way straight-line motion, the magnitude of average velocity equals the average speed. Because in one-way straight-line motion, distance = magnitude of displacement. We covered this earlier. Going straight, with no turning back, getting there fast means running fast. So, magnitude of average velocity equals average speed. Let's try another problem.

The earthquake rescue march

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Again, we get a description and must decide whether it means average or instantaneous, velocity or speed. This one: an earthquake rescue march. During an earthquake rescue, a rescuer walked 60 km in 5 hours. Is 3.3 m/s an average velocity, a velocity at every instant, an average speed, or a speed at every instant? Where does 3.3 come from? 60 km ÷ (5 × 3,600 s) is 3.3 m/s. So what is 3.3 m/s? It must be an average speed. Not average velocity: 60 km is distance, not displacement. He walked a rugged mountain road. Could that be dead straight? It winds, so 60 km is the distance. So it's speed, not velocity. Any option with velocity is wrong. Average, or at every instant? It never says the speed stays constant. It only gives total distance and time.

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So it's average speed. The answer is C. Once more: for average velocity, divide displacement by time. For average speed, use distance over time. That's all for average speed. Now, instantaneous velocity.

Instantaneous velocity

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Aren't average velocity and speed enough? Why study instantaneous velocity? Average velocity and speed only give a rough idea of motion. But I want more detail, especially for athletes. Take the famous Su Bingtian, the pride of Asian track and field. When he trains, he breaks the run down very finely. I want to know how my start went, and how it compares with before. How is my first stretch, my middle one, my final sprint? I want every second of the run in detail, or even over shorter times. Am I fast at the instant I start? At the instant I cross the line? I want very fine detail. So I record many instants of the run and how fast I am at each. If I only said that overall, you ran 100 m 0.01 s faster than before, how would I know where that came from? I wouldn't. Break it down, strengthen each part, then compare data to improve.

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It's just like studying. If you're doing badly in one subject, break that subject down further. Say you're bad at physics. Find out if it's linear motion or force analysis, etc. Only by breaking it down can you solve it. If you don't, you'll never solve them. Digression over. Back to the topic. Knowing the speed of the whole process isn't enough. Sometimes I need an object's speed and direction at one instant or place. That defines instantaneous velocity. Many can't grasp "an instant, a position." Isn't that just a point? A point should be at rest, so how can it have velocity? Just think of it this way: an instant is an extremely short time. That's fine: physics isn't math, it has to consider reality. An instant isn't really a point, as a point can't be measured.

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So an instant is an extremely short time. For example, this 0.0001. I picked it myself. To me, extremely short meant 0.0001 s. So you can think of instantaneous velocity as the average velocity over 0.0001 s. That's enough: the average velocity over an extremely short time. It's a vector too. But its direction differs from average's. Take this car, going from A to B. The average velocity over this stretch points along an arrow from A to B. Instantaneous velocity isn't like that. Take this spot. Its velocity points along the tangent, where the car's front points. Take this point, and the direction is this arrow. This point: this arrow. This point: this arrow. In curved motion, whichever point I take, the direction can differ.

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So its direction is defined differently from average velocity's. Never think they point the same way. One points from start to end, ignoring the process. The other follows the tangent, the way the car faces. Or think of running: the way your face points at an instant is the direction of your instantaneous velocity. It keeps changing as you turn. That's instantaneous velocity. With instantaneous velocity, we can define

Constant-velocity motion

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the simplest, most basic motion model in high school: constant-velocity motion. Constant-velocity motion is motion with constant instantaneous velocity. That means two things. First, the magnitude is constant. Second, the direction is constant. If velocity direction never changes, it moves one way only, so only in a straight line. Any turn changes direction, so no turning. And its speed is constant too, so it moves uniformly. Isn't that constant-velocity motion? Sometimes you get a true-or-false: "Constant-velocity motion: motion with constant instantaneous velocity?" Boldly choose true! It's exactly that kind of motion.

Average or instantaneous? Five cases

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Now, another problem on telling average from instantaneous. It tests if you truly understand it. instantaneous velocity is the average over an extremely short time, say 0.0001 s. To tell which one a description means, check if it's an extremely short time. That's all. First: in a 100 m race, an athlete crosses the line at 9.5 m/s. This describes the instant he crosses it. Think about it: is crossing an extremely short time? The nose touches the line, and he has crossed it. An extremely short time. So it's instantaneous velocity. Yes, it's an extremely short time. How short is extremely short? No fixed rule. Go by feel. Second: After the speed-up, a train hits 150 km/h. That's an average velocity.

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Why speed up a train? To reach the destination sooner. If it hit 150 km/h for only one instant, then dropped back down, arriving earlier wouldn't change. So we check whether the average velocity has increased. The greater the average velocity, the sooner I arrive. Hence average velocity. Third: a traffic jam, the car's speed in the tunnel is 1.2 m/s. That's an average velocity, right? It's calculated from the car going in to coming out, using the tunnel length (displacement) divided by the time. A car can't clear a tunnel instantly. So, average velocity too. It spans the whole time in the tunnel, which is a time interval. Fourth: a returning space capsule lands in the Pacific at 8 m/s. Once it first touches the water, we say it has landed.

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That's also an extremely short time. So it's also instantaneous. The last one: a bullet hits a wall at 800 m/s. Impact is also an extremely short time. So it's also instantaneous. After these five exercises, you should know how to tell them apart. Just look at the problem's description. An extremely short time: instantaneous. A longer time, like a car in a tunnel, is average. That's all for instantaneous velocity. Now let's look at instantaneous speed.

Instantaneous speed

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Instantaneous speed is simple: the magnitude of instantaneous velocity. Its meaning is like instantaneous velocity but it ignores direction. Drop direction: instantaneous speed. Simple as that. Careful: average speed is not the magnitude of average velocity. Instantaneous speed is the magnitude of instantaneous velocity. Keep them apart. A scalar too. That's it for the four core quantities. Now, comparing speed and velocity,

Comparing speed and velocity

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what to watch out for: First, velocity and speed can't be compared in size. Because velocity is a vector. As we said in vectors and scalars, a vector is an arrow, while a scalar like speed is a number. Arrows and numbers can't be compared. To compare them, you can only use the magnitude of velocity against speed. Again, we take instantaneous and average in turn. Instantaneous speed equals the magnitude of instantaneous velocity. That's right. By definition, it's the magnitude of instantaneous velocity, so they're equal. Many students don't dare pick 'equal'. Remember: they are equal. A vital idea in high-school physics, the basis of the 'tiny slices' idea. Take any motion, however complex its path may be, straight or curved. Now take a piece from the middle, an extremely short slice of time. When time is extremely short, is it straight or curved? About the same. So short that straight and curved paths are about equal.

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Changes in velocity can be ignored too. Over such a short time, asking if speed changes is pointless. So when I take an extremely short slice, I treat that motion, whatever the actual motion is like, as constant-velocity motion and analyze it. That's what the method rests on. Again: in an extremely short time, changes in direction, path and speed mean nothing: treat them as the same. So instantaneous speed equals the magnitude of instantaneous velocity. That's the instantaneous case. Now the average case. Average speed is always at least the magnitude of average velocity. Say I go from A to B. Any route: average velocity is displacement ÷ time. A straight line is the shortest of all paths. If I don't go straight, average speed is greater than the magnitude of average velocity. When does the equal sign hold? Again: one-way straight-line motion. There, distance equals the magnitude of displacement, so average speed equals the magnitude of average velocity.

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That covers comparing speed and velocity. Now two more practice problems. First:

Practice: one lap on a 400 m track

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At a school sports meet, Teacher Liu runs a lap of a 400 m circular track, back at start in 80 s. What was his average velocity? Note: he got back to the start, so the average velocity is 0. No change in position. And his average speed? Use distance divided by time: 400 divided by 80 is 5 m/s. As Teacher Liu crosses the finish line, is the magnitude of his velocity equal to his speed then? Note: 'crossing the finish line' means an extremely short time. So the velocity at the finish line is the instantaneous velocity, and the speed is instantaneous speed. So the magnitude of instantaneous velocity and speed: how are they related? They're equal, as we said. Comprehensive: it tests several points, especially the last sentence: Translate the wording accurately into what it really says. It takes an accurate, deep grasp of the physics concepts.

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As I told you earlier: to tell instantaneous from average, look at the description. An extremely short time: instantaneous. A not-so-short time: average. This helps a lot when solving problems. One more multiple-choice question.

Practice: three particles from N to M

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Particles A, B and C take the paths shown: A is red, B is blue, C is green. All leave N together, reach M together. Which statement is correct? A. From N to M, A has the greatest average velocity. For average velocity, ignore the path; look only at displacement. Same displacement from N to M, same time, so all average velocities are equal. A is wrong. B, same average speed? Wrong too. A and B are swapped: average velocity same, average speed not. C: A has the greatest average speed. Average speed is distance divided by time. All three take the same time, A's distance is greatest, so A's average speed is greatest. C is correct. D. On reaching M, A has the greatest instantaneous speed. Is D correct? Wrong. Why? Because we can't tell. The data don't give instantaneous speed. We only know each one's total distance, displacement and time.

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Is speed highest at M, midway, or at the start? Unknown. So D must be wrong. The answer is C.

Comparing all four

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Finally, let's put the four quantities together and compare them. To understand average and instantaneous, you need to know two statements: Average speed is not the magnitude of average velocity. Different meanings: speed: how fast you cover ground, velocity: how fast you get there. But instantaneous is different. Instantaneous speed is the magnitude of instantaneous velocity. Similar meaning; one has direction, the other doesn't. Very simple. Now this diagram. Summarize it yourselves: Shrink average velocity's time interval to extremely short: instantaneous velocity. Likewise, shrink average speed's time to extremely short: instantaneous speed. But is the magnitude of average velocity the same as average speed? No: less than or equal to average speed. That comes from their different meanings. When is it equal?

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One-way straight-line motion. Here, running fast is arriving fast, there's only one path. So the magnitude of average velocity equals average speed. For instantaneous, no such split: as it's an extremely short time. I ignore the actual trajectory, and velocity changes too. So the magnitude of instantaneous velocity equals instantaneous speed. Grasp average versus instantaneous, and velocity versus speed, and you know velocity thoroughly. One last problem, true or false.

Five true-or-false statements

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Judge each statement as true or false. First: instantaneous velocity is the average velocity of a moving object over a short time interval. Note: it's an extremely short time, not just short. So this is wrong. Change 'short' to 'extremely short' and it's correct. 'Short' is not 'extremely short'. Second: for constant-velocity motion, average velocity doesn't depend on the time interval or displacement. Correct: in constant-velocity motion, whatever time interval you take, you get the same result; nothing has changed. So it doesn't depend on the interval. Third: instantaneous speed is the magnitude of instantaneous velocity. This is correct. Countless students were afraid to pick it in their first year. But after this video, you can pick 'true' without hesitation. By definition, instantaneous speed is the magnitude of instantaneous velocity. Fourth: average speed is the magnitude of average velocity?

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Wrong, I've said so often. Fifth: average velocity zero, average speed zero too? Not so! Take one lap: average velocity is zero. Average speed? Not zero: displacement is 0, distance isn't. One last small point to explain. We've covered four quantities,

Which one does "sudu" (速度) mean?

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so you may wonder: which 'sudu' do we mean in daily life? The textbook already gives the answer. Here's a textbook screenshot. Look at the last sentence: In Chinese, the one word 'sudu' sometimes means speed. Judge from context. Of the four quantities, average velocity, average speed, instantaneous velocity and speed, all get loosely called 'sudu'. Which one is meant depends on the context. In Chinese problems, a bare 'sudu' is read as instantaneous velocity. Sometimes it's the average velocity of constant-velocity motion, equal to instantaneous velocity. So in Chinese problems, 'sudu' alone is instantaneous. In English, check speed/velocity.

Summary

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There was a lot today, but only two big things to master. First, the four quantities: average speed and average velocity differ in physical meaning. Many differences, but instantaneous speed and velocity are similar in meaning. So the magnitude of instantaneous velocity is instantaneous speed. Screenshot this four-quantity chart. Second: constant-velocity motion: motion with constant instantaneous velocity.