How to Find Change in Velocity: Why It's 7, Not 3

Cold open: Δv is 7, not 3

0:00
Final velocity vt equals negative 2 m/s, since it points opposite to positive. Then Δv is negative 2 minus 5, negative 7 m/s. So Δv has a magnitude of 7 m/s. Direction? It's negative, Positive is down, and it's opposite, so it points up.

Why this lesson: the setup for acceleration

0:25
This lesson is on change in velocity. Its main goal: set up acceleration, coming next lesson. Acceleration is among the most important physical quantities in high school. Many never get acceleration and are lost for 3 years. So these two lessons lay the groundwork. We'll nail the core idea of acceleration. Before change in velocity, I want to cover change itself.

What a change means: from initial to final

1:03
99% of teachers go straight to change in velocity and rush past it, thinking there's no need to explain in detail. That makes later topics much harder. So today we settle this properly. What is a change? It describes how something goes from its initial state to its final state. Let me say it again: a change describes going from initial to final. Here's an example. Last term I scored 50 on my exam, 50 in physics. I'll call that my initial state. Then I found a teacher, Tiaotiao, and took his course. Next term I scored 90. I'll call that my final state. So from 50 to 90, the change in my score describes how much I changed from 50 to 90, and whether it went up or down.

2:17
From 50 to 90, my score changed by 40 points. Did it go up by 40 or down by 40? Obviously up. Up by 40 points is the change from 50 to 90. So what is a change for? It shows how we go from initial to final. It carries two pieces of info. First, how much it changed. Second, whether it increased or decreased. It must carry both. Every quantity has a change we can calculate. Quantities are scalars or vectors. A vector has magnitude and direction. A scalar is just a number, with magnitude only. Scalar changes are simpler, so let's study those first.

Scalar change: final minus initial

3:21
Here are two scenarios. Let's calculate the change in each. First: I work hard for a month and earn some money. My balance goes from 3000 to 4000. What's the change? First, it increased. It's an increase, not a decrease. As I said on the last page, a change has two pieces of info: up or down, and by how much. Here it went up, so I use a plus sign for an increase. By how much? From 3000 to 4000, that's up 1000 yuan. So the change is +1000 yuan, an increase of 1000 yuan. Second example: you're watching my video. You got it. Great! and tip me some coins. Tipping uses up your coins, right? Your coins drop from 100 to 98. What's the change in your coins?

4:27
It decreased, right? So I use a minus sign for a decrease. How many? Two: so minus 2 coins. That's the change in your coins. Let's summarize. When calculating a change, the sign shows increase or decrease: plus means increase, minus decrease. The number shows the amount. Look at +1000, 3000 and 4000. 4000 minus 3000 is +1000. In the second, 98 minus 100 is -2. So a scalar's change is simple: final minus initial. If final is larger, you get a positive. If final is smaller, it's negative: so a decrease. Note: a change is final minus initial.

5:33
Never initial minus final; the sign would be backwards. We use final minus initial, so the sign tells increase or decrease. That's how we calculate a change. Pretty easy so far, right? Now I'll set change aside for a while.

"Depends on" vs "can be calculated"

5:53
I want to cover something very important in high school physics. This is a full course, and once these are done, later is easy. Many teachers skip this. It's not hard, but because they skip it, it troubles students for three years. What's the problem? How do I tell if two quantities are related? We often get questions asking if this depends on that, and whether an option can be chosen. Many students find this a headache. Not hard, but nobody taught the analysis. First, note this. We must separate two things: "related" and "can calculate". Countless students share one mistake. What mistake? Thinking that quantities in one equation are all related. For example, v equals s over t. Speed equals distance over time.

6:59
These three appear in one equation, so all are related. Remember this. If quantities share an equation, you can only say I can calculate or analyze it this way. Take v equals s over t. They're in one equation; you can only say speed can be calculated from distance and time. That's "can calculate". But "can calculate" isn't "related". So what does "related" mean? Know who relates to whom. "I depend on you" means: if all other conditions stay the same and you change, I must change too. Only then can I say I depend on you. You: independent variable. Me: dependent. Now apply this to the equation.

8:06
Does speed depend on distance? No. Say a car moves at constant speed. The farther it goes, the more distance, so distance changes, right? But when distance changes, does speed change? No, because its speed is constant. Distance changed, speed didn't, So no: speed does not depend on distance. Got it? Even in one equation, they may not be related. You can only say speed is calculated but speed doesn't depend on distance. So how do I tell if they're related? Use the physical meaning. Take the same example: a car on the road. You said its speed doesn't depend on distance or time.

9:11
Depends on what, then? On how good the car's engine is, and how hard the driver hits the gas. If the driver floors it, my speed is surely higher, right? A better, stronger engine? I go faster. So speed depends on these factors. We analyze it through physical meaning. Speed depends on the driver, the engine, the gas pedal, and how you drive. It doesn't depend on distance and time. Note this.

"Depends on" goes one way only

9:48
Next: "related" is one-way. What does one-way mean? Let's use the same example, Speed v: v equals distance over time. Here, my distance depends on speed. Suppose time is fixed: one hour each. The greater my average speed, the farther I can go. So my distance depends on speed. If I floor the gas or get a better engine, My speed goes up, my distance follows. Distance is affected by speed. But it doesn't work in reverse. Distance depends on speed, so speed does not depend on distance. If you depend on me, it means I affect you, but you can't affect me back. Distance is affected by speed, but distance can't affect speed.

10:53
Speed is affected by engine and gas pedal. It's like this. You watch my videos closely and your grades go up. Your grades improve. That depends on my videos. My videos affected you. But in reverse, can your grades affect my videos? No way, right? So in physics, "related" is always one-way. If A depends on B, then B does not depend on A. Think of this example about distance and speed: Distance depends on speed, not vice versa.

Change vs initial and final state

11:34
Now back to the change we covered. Change can be written as delta, which equals final minus initial. So many students think change depends on final and initial. Remember that misconception? They can be written in one equation. You can only say change can be calculated as final minus initial. But does change depend on both? I'll use an example again. Let my money in my pocket today be my initial state. Then the same day next month, my pocket money is the final state. So the change is how much my money changes from this month to next. First, let's analyze the change and the initial state: are the two quantities related?

12:37
Note: they don't depend on each other. The money in my pocket today is determined by my past, right? Say I won the lottery yesterday, or Xiaoyu kindly gave me 100 yuan. Past events affect my money today, right? But my change depends on what happens today through next month, this time interval. Did I earn well? Did I lose money? Or did Qi Xiaoyu give me some? These two are independent. Got it so far? Change and initial state are independent. Change depends on my future earning power and my experiences, or any lucky break. My initial state is set by the past. First, they're independent. Now, the final state, note this: Final state depends on initial state and change.

13:44
Final state can be written as initial state plus the change. If the change is positive, final is higher; if negative, it's lower. I can just use addition, adding a positive or a negative. Look, say I earn a fixed amount. 1000 yuan a month. If the change is fixed, the more money I have right now, the more I'll have at the end, right? If my initial money is fixed, and I earn faster, so the change is bigger or smaller, the final state changes too. So, how much money I'll have next month depends on the money I have today and whether I earned or spent in between. Both factors affect next month's money. So final state depends on both of them.

14:54
This relationship is one-way. Since the final state is affected by the change, it can't affect the change back, right? Change isn't set by the final state alone. Not by initial or final state alone. The change is set by some outside factor. It doesn't depend on either state alone. Back to our coin example. Real examples really help with physics. It's my top method. I love examples. Look, say you give coins: After giving coins, does the coin change depend on your start or end count? No. What does it depend on? Whether you press that button. It's up to the platform's rules: 2 coins or 3 coins.

16:03
Not your coin count before or after. Note this point. On change, here is the key point, the key point of this lesson. I hope you fully understand that change doesn't depend on either state alone. It's set by extra outside factors. Next: change in velocity.

Velocity change is an arrow

16:37
Velocity is trickier than money and coins because it's a vector with a direction. Let's go step by step. First, let's split the change into an increase and a decrease. Remember vectors? Each is an arrow. Velocity is no exception. Say an object moves to the right, with velocity 3 m/s. That's a right-pointing arrow of length 3. It means a velocity of 3 m/s to the right. First, the increase. Say my velocity increases by 1 m/s. How do I show that increase of 1 m/s? If velocity increases by 1 m/s, what does it become? It becomes a velocity to the right, of 4 m/s, right?

17:48
If initial velocity is 3 m/s, plus 1 m/s, the final velocity must be 4 m/s. The change must show how I get from right 3 to right 4. Do I just add a small arrow? Look, I add a little arrow here. Two vectors on the same line: same direction: add opposite: subtract. Take a right arrow of length 3, add a right arrow of length 1: you get a right arrow of 4. So when velocity increases by 1 m/s, the increase is also an arrow, pointing right, with length 1. Joined to the initial, it gives the final. So the increase is an arrow too. Now, the decrease.

18:54
How do I show a decrease of 1 m/s? If velocity decreases by 1 m/s, the final velocity becomes 2 m/s to the right. How do I describe going from 3 to 2? Just cancel out a small piece of arrow. Look, now my final arrow is shorter. Cut off a bit: initial becomes final. So decreasing by 1 m/s is an arrow pointing left. That's the increase. And the decrease, also an arrow. Together they make up the change, right? So note: the change in velocity is also an arrow. It must show speeding up or slowing down, and by how much.

20:04
Earlier with scalars, a plus sign meant increase, minus meant decrease. Vectors are a bit trickier. If the arrow of the change points the same as initial velocity, it's an increase: the arrow gets longer, Same direction adds up. A longer arrow means velocity increased: speeding up. If the change arrow points opposite to initial velocity, velocity decreases, since opposites subtract. So to tell speeding up from slowing down, it's trickier now. Compare change arrow with initial arrow: Same: speeding up. Opposite: slowing down. That's how we tell increase from decrease. How much did it change? Easy: just read the arrow's length. That's the change in velocity.

21:12
Next, let's draw arrows in a few cases.

Three cases: speed up, slow down, reverse

21:14
Three scenarios. Let's go one by one. Then you'll be fine. Group 1: initial is 3 m/s to the right, final is 4 m/s to the right. So it's speeding up to the right, right? Since it's speeding up, the change points the same way as initial; The change is first of all a right arrow. By how much? From 3 to 4: up by 1. So it's 1 m/s pointing right, the red arrow. Group 2: initial is 3 m/s to the right, final is 4 m/s to the left. What does that mean? Like when a driver takes you out, halfway you find you took the wrong road; going right, but you should go left.

22:20
It's actually two segments. First, hit the brakes, stop, then turn around and go left. Right to left: split it into two segments. Segment 1: from 3 m/s to the right, slow down to zero. Then from zero, speed up left to 4 m/s. We analyze the change segment by segment. Segment 1: from 3 m/s right, slow to zero. Remember? If it's slowing down, the change points opposite to initial velocity. This is the red arrow; it shows segment 1's slowing down, 3 m/s to the left. At zero, I start speeding up to the left. The change keeps pointing left: 4 m/s, right?

23:25
Two segments: 3 m/s to the left first, then 4 m/s to the left. Put left 3 and left 4 together: That's 7 to the left, right? So group 2's change in velocity is 7 m/s to the left. It has two stages: slow to zero, then speed up backward. Third: simple slowing down, from 4 to 3. Covered before, right? If it slows down, the change in velocity is opposite to initial velocity: leftward. How much? From 4 to 3, down by 1, so 1 m/s to the left. Got how to analyze change in velocity? These three cases cover everything: one speeding up, one slowing down, one slowing to zero, then reversing. Just three cases. Got it? Let me stress again: the magnitude and direction of change in velocity isn't set by v0 or vt alone.

24:31
Its direction depends on whether you hit the brake or the gas. Hit the brake, it's opposite to initial velocity. Hit the gas: same direction as initial. Direction depends on brake or gas, and magnitude on how hard you press. Harder press, bigger change, right? Note: change in velocity doesn't depend on v0 or vt alone. Not on initial or final velocity alone. I analyzed this in detail earlier. How change in velocity relates to v0, vt is often tested in intro kinematics. If it's tested, remember: Magnitude and direction: not set by v0, vt alone. It's decided by me, the driver, by whether I hit the brake or the gas, and how hard I press not by initial or final velocity alone. Now recall something.

On a line: plus and minus signs

25:31
When we covered displacement, displacement was an arrow, a vector? In straight-line motion, we simplified displacement with a trick: turning an arrow into a signed number. How? We set a positive direction. One way positive, the other negative. Same as positive direction: positive. Opposite: negative, plus a number. Easier than drawing arrows, right? We can do this with change in velocity too, but only for straight-line motion. If an object moves in a straight line, we can simplify the notation for Δv. Step 1: set a positive direction. Same example as before. Take right as the positive direction. The initial velocity is positive, 3 m/s, because it points right.

26:36
Plus sign can be dropped. Final velocity: 4 m/s. The change in velocity, as analyzed: 1 m/s to the right. We can write it as +1 m/s. Right, same as positive: positive. Magnitude 1, so the change is +1 m/s. Group 2, same way. Final velocity is opposite to positive, so the final velocity is -4 m/s. And the change in velocity, as before, is 7 m/s to the left. Left is opposite to positive, so write it as -7 m/s. Group 3: the change in velocity is 1 m/s to the left, so it's -1 m/s. Same method: a sign plus a number. From these three examples, we get a rule. What rule? Note the plus/minus sign shows direction, not the math kind of plus and minus.

27:48
But if, in calculating, we treat it as math signs, and use final minus initial, we get what the arrow method gave. Take group 1. The + here isn't a positive number, it just means same direction, but we treat it as math and use 4 - 3, to get +1, the same as before. Group 2: Here -4 only means direction left, but we treat it as an ordinary -4. -4 minus 3 gives -7, same as the arrow result, right? Group 3, same: Use 3 - 4 and you get -1, Same as the arrow result, right? Once we set a positive direction, and turn v0 and vt into signed numbers, we can directly use final minus initial, an algebraic way, to calculate.

28:54
Note: we only calculate it this way, but the sign still means direction, not math's positive and negative. It has nothing to do with >0 or <0. It just happens to give the same result. So for straight-line motion, to find the change in velocity, treat signs as math: final minus initial, and the result is the change in velocity. That's it for this part. Now let's summarize. Straight-line change in velocity is key,

Recap: positive direction, then v_t − v_0

29:36
and it's the basis for acceleration later. To find change in velocity in a line, Step 1: set the positive direction. If you don't set one, the signs have no meaning. You must say what positive means. Next, determine the signs of initial velocity v0 and final velocity vt. Same as positive: +; opposite: -. Step 2: just use vt - v0 to find Δv. Treat them as ordinary numbers. Plug in the signs and compute. Δv is also a vector. Positive result: positive direction, Negative: negative direction. Note: this method works only in straight-line motion. Because it has just two directions, On a line, one way is +, the other -, so two signs cover all directions.

30:47
What about a curved path? Look: a direction here, and here, and here... infinitely many. Signs alone can't describe it, Only straight lines work. Note: in calculation, velocity's sign is used as a math sign. One last point, a reminder: magnitude and direction of Δv aren't set by v0 or vt alone. I've stressed this more than once. It's important, so one last time. Note this. That's all the knowledge for today. Now a few problems to practice.

Practice: compute Δv for three motorcycles

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First, a change-in-velocity exercise. Motorcycles, all starting to the right, find the change in velocity in all three. Step 1: set the positive direction. I suggest making the initial velocity's direction positive. How to choose it? Up to you, anything works. Here, left or right both work. But most teachers' habit when explaining is initial direction positive, me too. And in most workbooks, answer keys, too, take initial direction as positive. So stay consistent with them; it's easier. So I take right as positive, to match. Group 1. Initial: 30 km/h to the right, so +30. Speeds up to 100, surely to the right, so final is +100. So Δv is just 100 minus 30, treating them as ordinary numbers.

32:38
That should be 70 km/h. So the change in velocity is 70 km/h. No need to say left or right, since + already means right. The + can be dropped; the - cannot. Group 2: slows from 80 to 20. The 80 is to the right, so +80. 20 is also right, so final is +20. So Δv is 20 minus 80, Final minus initial gives -60 kilometers per hour. Group three: I start at +40, to the right I brake, turn around, to 20. I turned around, so that 20 points left so it's negative 20. So Δv is negative 20 minus 40, which is negative 60 Units: kilometers per hour. Practice again: how to compute this? Practice well. Next lesson, to find acceleration, we need Δv first.

33:46
So acceleration builds directly on this. Make sure you master it in this lesson. Easy, just a few steps. Step 1: set positive direction Step two: signs of v0 and vt. Step three: just subtract. Practice just this point.

True or false: what Δv does not depend on

34:04
Next: more concept questions to reinforce the concepts. Honestly, I explained earlier in detail, so these problems feel easy now. Look, number one: Velocity is a vector, Δv is a scalar. Wrong. I said it's an arrow, a vector. Second: velocity points right, the change in velocity may point left. Correct! Its direction does not depend on velocity. That's initial velocity. Say a car moves right. If you hit the gas, it speeds up; Δv points right. You brake, it slows, Δv points left. Whether Δv points right or left depends on gas or brake, not on your initial direction. Got it? So the second is right. Third: higher velocity, means bigger change in velocity? Nonsense, right? Is a bullet train fast?

35:06
Very fast, high speed. But velocity is constant, so Δv is zero. No relation. Magnitude and direction of Δv come from outside factors, not velocity. Third is wrong. Fourth: Δv always points along final velocity Wrong, no dependence. The last one: The bigger Δv, the bigger the velocity? Here, velocity means final velocity. Is it always larger? No. What if it slows down? When slowing, a bigger change means a smaller final velocity. Not right either. The key point of this group: What I most want you to fully understand: the magnitude and direction of Δv do not depend on v0 or vt alone. Outside factors decide it. Think: brake or gas pedal. Okay, that's all I'll say about this one. Next, Problem 3:

Multiple choice: which statements about Δv are right

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Which statement about Δv is correct? A: Δv is a vector, no problem. In a line, signs show direction. We covered that, right? A bigger Δv means a bigger velocity? B is wrong. C: if final velocity points right, Δv may point left. Note it says 'may'. So C is correct. It could go left or right. D: Δv shows how fast velocity changes Careful: that's wrong. It shows how much velocity changes: the amount. How fast it changes is acceleration, coming next lesson. So don't dwell on D; it's wrong. Answer: A and C. We'll cover this in detail next time.

Example 4: the bouncing ball

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No more on this one. Next, two calculation problems. A ball slams down onto the ground at 5 m/s So its initial velocity is 5 m/s downward. It bounces back up. After bouncing, while it rises, At one instant its speed is 2 m/s, so the final velocity is 2 m/s upward. That's initial, that's final. Find the magnitude and direction of Δv. Follow the steps as usual. Step one: choose the positive direction: down, the initial direction. Initial velocity v0 equals 5 m/s. Final velocity vt equals negative 2 m/s, since it points opposite to positive. Then Δv is negative 2 minus 5, negative 7 m/s.

37:58
So Δv has a magnitude of 7 m/s. Direction? It's negative, Positive is down, and it's opposite, so it points up. The answer is A and D. Now, the last problem this lesson.

Only speeds given: two possible answers

38:15
It's a bit harder. It's tricky: there may be several cases. Let's take a look. A particle moves in a straight line, magnitude of initial velocity 2 m/s, magnitude of final velocity 4 m/s. Find the change in velocity. Note this: only speeds are given, not direction. That leads to several possibilities. Again, take the initial direction as the positive direction. Whichever way it goes, it's positive. So initial velocity v0 equals 2 m/s. Whatever direction, v0 is positive. That's my rule, okay? Rules are rules! And the final velocity? I only know its magnitude: 4 m/s. It moves in a line, so its direction has two options: same or opposite.

39:22
So I can write vt as positive or negative 4 m/s, it could be either one. So Δv has two possibilities. Case one: vt is +4 m/s, so Δv is 4 minus 2, which is 2 m/s. Its direction: same as initial velocity, since it's positive. Case two: final velocity is negative 4, Δv is negative 4 minus 2: negative 6 m/s. Its direction matches the final velocity, opposite to positive. Let's check the options. The magnitude could be 6, or 2, 4 can't happen, right? So A is correct. Direction: positive or negative, it may match the positive direction. Same as positive is positive, opposite is negative.

40:26
Either is possible. With two cases, C and D are both right. So the answer is ACD. Note: if only magnitudes are given, with no direction, stay alert: there may be several cases. Keep this in mind. This lesson isn't hard, but there's a lot to understand, so it may fry your brain a little. But it matters well beyond this early stage; it matters through all of high school. So please watch it several more times and fully master it. It's not hard: I gave many examples. With examples, it's easy to follow.

Summary

41:10
First: the change in velocity. Note: for each new physical quantity, first know roughly what it looks like. Change in velocity looks like an arrow. If the arrow is along initial velocity, it means speeding up: velocity increased. If opposite, velocity decreased. The arrow's length shows the amount. Also, its magnitude and direction aren't set by initial or final velocity alone. They're set by outside factors. It depends on gas pedal or brake. Second: given a straight-line motion, find the change in velocity. Step one: set the positive direction. Then compute vt minus v0. Treat the signs as math plus and minus, and just calculate.

42:13
If it comes out positive, Δv points positive; negative means negative.