Okay. So in this chapter we are going to discuss about motion in one dimension. Right? So I think many of you have seen this in high school if you took high school physics and so this belongs to I mean this this material correspond to section 2.1 of our book and again we are using the 10th edition right? If you purchase 11th edition, if I mean if you ever purchase hard copy of the 11th edition, the chapter number are different, right? The the chapter we call it chapter two, they call it chapter three in the next edition. But currently we stay with the 10th edition. So it's chapter two, section one. So section one, we consider the definitions of positions, displacement, velocities and speed, right? And so on. So first position, right? So when you try when we try to analyze motion of an object very often like the textbook authors we treat it as a each object we can treat it as a particle. For example this car here when car moves from one location to another you can treat car as a particle but you may wonder why I mean can can we really treat car as a particle car is pretty big right? So when we treat something as a particle it we didn't mean that whether it's big or small. When we treat something as a particle we mean that we can ignore the irrelevant degrees of freedom. For example a car consider a car here right? A vehicle or a car there are four wheels right? So you know that each wheel is rotating is rotating here. So by treating car as a particle we mean that we can ignore the rotations of the wheels those are irrelevant degrees of freedom right? So you can if you consider car as a particle. So particle if you treat an object like a car as a particle we mean by treating it as a particle we mean we can we can ignore okay or disre we can treat this as a particle by disregarding the so-called irrelevant degrees of freedom. So what we mean by irrelevant degrees of freedoms is for example at this point I I'm not really interested in analyzing the rotational motion of the the wheels. So that that means if I'm not interested in I can treat the whole car as a particle and those irrelevant rotational degrees of freedom will be suppressed right and we just consider the translational motion of the car as a whole object as a point and that point is moving from one location into another and so on. Right? So, so first you need to understand that in the test book they treat something as a particle mean means the following right by we can consider some object as a particle as a point by disregarding all other irrelevant degrees of freedom and we can focus on translational motion of the vehicle here. Right? So now the next thing we would like to define is called displacement. Right? So in chapter two here and section 2.1 the first I mean one first of course first thing you define position but the position would just mean location of a car and then now you can next thing you can define is displacement. So displacement of an object now we have treated as a particle just a point right. So that means you can define your coordinate system X coordinate and this is your origin O right and at some point say at t equals z suppose the object was somewhere somewhere here say t equals z the object was here go x0 but the next moment t let's say t = 1 say 1 seconds the the object is located on x1 right and t = 2 seconds the object is located at x2 2 right and let's say t equals 3 seconds the object come to here right so this let's say this is x3 when t equals 3 seconds and so on so now if you can consider the position the evolution of position as a function of time then you can define something called displacement then displacement is defined usually defined as delta x this delta means ch a change a change in position is called displacement And displacement is defined as xfal minus x initial. Right? So that means displacement means the position at a later time minus uh position the position of the particle at a former time. Right? So this final doesn't mean the ultimate final just you pick two time uh two time instance and the later time position minus the former time position that will give you displacement. For example, if you look at this thing here x1 x2, right? And if you consider the displacement from t = 1 second to t = 2 seconds and that will give you this thing, right? X. So for example the from X1 to X say delta X from T to uh from from t equals uh t goes from t equals 1 to 2 seconds. Okay 1 seconds to 2 seconds. For instance if you want to compute this delta x then you will get x2 minus one like this. Okay. So this is simple stuff. Actually displacement is actually a vector actually. So basically you when you consider x1 here so that's your here. So this is your x1 and this is your x2 position x1 position x2 position and you consider x2 minus x1. So your displacement here is represented by this vector here. So this is your delta x vector. Okay. So let me say again. Okay. So this x can be considered as a vector in one dimension. Your x1 is the position vector in one dimension. x2 is the position vector in two dimensions. And when you do a subtraction of x2 - x1, you get the third vector. It's a displacement vector. Displacement vector. So that means when when this particle when time goes from 1 second into 2 seconds and this particle will travel from this location to this location here. Right? So in this case you can see that this vector points toward the right. Right? But now if you consider the relation between x2 and x3 then you will see that the displacement um the displacement vector you you can consider that as well right and that vector will points toward the left actually right consider but we don't go into too much detail right now because when you work on the homework you you'll see the meaning of displacement this displacement can either be positive or negative right displacement can either be positive or negative positive means that that vector vector components point toward the right and if you get a negative displacement that means that that vector components points toward left. Right? So once you have defined displacement we can go ahead to define the so-al average velocity. What is the average velocity of a car? Right? So now we treat a car as a particle and try to define what is the average velocity of a car when when t let's say when t goes from evolves from 1 second into 2 seconds for example right then how do you define a so-al average velocity so now let's consider this as follows average velocity sorry let me let me write it here let me write it here average velocity We know that velocity is defined as time rate of change of position right this kind of average velocity. So if you consider this average I say V our textbook notation call it V subX right and they use FVG but sometimes people can use a bar to denote an average okay let's use a bar here if you want to consider average velocity can be considered as the time rate of change of dis the position right so you can write it as delta x over delta t and then you can write it as displacement means the final position minus the initial one right and then this is the time interval for the displacement. So that means this is t final minus t initial. Right? So now now I can we can give you example for example for instance if you would would like to consider the average velocity from t1 to t2. So if you consider the say t goes let's say I use a mathematical notation let's say t would goes from t1 to t2 okay from t1 to t2 here right so this this uh means this notation okay this means belongs to right so if t goes from t1 to t2 then how would you define this average velocity in terms of this time interval right because for each average velocity you should specify your time interval So that means from t1 to t2 the average velocity v xar can be defined as uh x2 minus x1 and t2 minus t1 and so on. So that means average velocity you should specify which time interval you are interested in. Right? Say from 1 second to two seconds or from two seconds into three seconds and so on. Right? So again since average velocity is the ratio of displacement to time time can never be negative right the time interval can never be negative we don't have negative time in nature but displacement can either be positive or negative you you know that right oh I'm sorry I should change the battery let's move on okay so now let's proceed from here so let me say again so this is the general general expression. But for each example, you should specify the time interval first, right? Say from 1 second to two seconds. Then you can compute the so-called average velocity within that time interval, right? Very soon, we'll see an example. Now, let's move on to the to the next. And what's the geometrical meaning of average velocity, right? That's something you can think about. So, think about this graph here. This is a so-cal XT graph. The vertical axis is denoted by X. Okay, position horizontal axis is time. Right? So suppose a particle is traveling in one dimensional space and the X versus time position versus timing graph look like this curve. Right? Then if I would like to get the average velocity of the particle between A and B, right? Between A and B, then I can use a formula, right? Use a formula. What is average velocity? Delta X over delta T, right? And then let me ask you this question. Between A and B, is the average velocity positive or negative? If you think it's positive, please please raise your hand. Positive. Hey, great. If you think it's negative, please raise your hand. Almost no one. No, no one. Uh, okay. So, but that's great. So, that's a good answer. Hey, between A and B, average velocity is positive. Now let's imagine between B and C. I I the next question I want you to compute the average velocity between B and C. This figure is in the textbook. Okay, you can read the test after the lecture. I can also upload the slides as well. But basically uh when I explain about things I the things written on the whiteboard are more important, right? But of course slides are also helpful between B and C. If I compute average velocity between the in the time interval between B and C. If you think the average velocity is positive between B and C, please raise your hand. Passive. If you think it's negative, please raise your hand. Negative, right? So, it's actually negative. So, think about this. So, if you can discuss after class why why is the average velocity between B and C negative? You can try to input some number. You will know that hey just consider x say x c minus xb and divided by t c minus tb and you will see that it's a negative quantity actually how about between b and d oh I'm sorry between let's say between c and d between c and d is the average velocity positive positive or negative if you think it's positive please raise your hand between c and d if you think that average velocity between c and d is negative please raise your hand Okay, great. So now you can see that when you have the concept, right, you don't really need to go through every mathematical details, right? Once you have the concepts, now you can generalize your argument to consider what's the average velocity between D and E and E and F, right? So then what what's the key point? Average velocity actually will give you the slope of the line segment, right? Say that if you look at the green line segment between A and B, right? That slope of that green line segment is passive. So the average velocity is passive. Right? Because average velocity is if you look at the graph is like the rice over run. Right? Right. Delta x over delta t. So the slope in the segment between a and b the slope of the green line is passive. So the average velocity is positive between b and c. The slope of that line will be negative. Right? So that's why average velocity is negative. That's how you observe the graph. Right? So very good. Right. So I think most of you got the point and and if you read the textbook you will see some numerical examples but we don't really need to go through those things. Right now let's consider another quantity called average speed. Average speed is actually very different from average velocity is defined as the defi as the distance traveled or total distance traveled over the total time required. So basically the numerical example you don't need to write down anything. Those are from the textbook, right? You just need to write down the formula. So, let me say again, average speed. Now, after this, we'll give you some some practice questions to think about, right? What's the difference between average velocity and average speed? Today, the time is short. Of course, on Wednesday, we we'll continue our discussion. Wednesday, we'll have more time. Average speed is a different quantity from average velocity. So let let's denote average speed by a different notation. Let's call it V subs and give it a bar. Okay, this is just my notation and it'll be written as total distance traveled over the total time required. So so this thing d is the total distance. So that means this qu this part is always positive and this part delta t is the total time required right total time. So that's what we meant by average speed right. So now let me give you uh one example since time is limited we just already go into one example here it's it's in the end of chapter problem is is actually in the homework problem but not really in the textbook example. Okay. So let's start with this one. Uh sorry this one. Uh this let's look at this second one here. Okay. Think about this example here. Sorry the phones are small. Let me explain. So there are two locations A and B. A person walks first at a constant speed from A to B and then that person travels backward from B to A at a different constant speed. Right? So now you need to obtain the average speed over the entire trip for the person and also you need to obtain the average velocity of the entire trip of the person. So yes, what's the question? Yes. Uh yeah, if it's not zero, it will be positive quantity. It will never be negative. Of course, can can still be zero, but but yeah. Yes. Okay. So do you understand this question? I'm sorry the phones are not not very big but so now let's think about this question here part A and part B the answers are different let's part B is actually simpler what's the answer to part B of the problem average velocity okay so average velocity how would you define the let me put that in this way let's let's work on part B first part B So let's consider a trip again. Okay. So this is location A, right? And location B. The distance is unknown, right? But it's fine. Let's call the distance L here. So the person travels from A to B with let's say with a say velocity V1, right? And what is V1 here? V1 is given 5.4. four 5.4 4 m/s and then once it reaches B it travels back from B to A with a different velocity right so let's let's just write down the speed the speed write it as a speed and it's like 3.3 right so now now let's consider part this question practice question uh 2.1 point B this question here let's answer part B of a question What is the average average velocity? Average velocity for the entire tree. Pay attention. Entire tree means from A to B then back to A. Right? So that means A is the initial location but A is also the final location. So what is the average? What is the average velocity for the entire trip? You should use the x a minus x a then final time minus zero. Right? And what is this? Zero. So average velocity is zero. Okay. So that's the answer to part B of the question. Let's look at average speed. Average speed means total distance travel over total time required. So total distance what's assume that the distance between A and B is L right. So then what is the time required for the forward trip? The speed is this one. Right? And distance is unknown L. So the time for the forward trip let's call that T1. And what is T1? How would you get that T1 or delta T1? The time required for the forward trip T1 is actually L over V1. Right? V1 is known but L is unknown here. So that's the you can call either call it T1 or delta T1. Let's say let's call it delta delta T1 for example or T1 is this one. And T1 is this. And what is T2 for the backward trip? T2 is the same L over V2. Right? So now we wonder what's the average speed for the entire trip. How do we compute? We should use the total distance over the total time required. Right? So then the average speed for the entire trip will be total what is total distance? Total distance is L + L 2 L over the total time required is this plus this L over T1. So sorry L over V1 plus L over V2. Right? T1 plus T2 and then you will see a cancellation of L. Right? You can see this. Then you will see 2 L over this L * 1 / V1 + 1 / V2. Right? And you'll see a cancellation of L. So you don't know L but it's fine. Right? Then you can get the answer very quickly. Right? So what's the answer? This answer you can find common denominator and then do the math, right? do the math then you can say that okay this is 2 V1 V2 over V2 plus V1 something like this but you don't need to get the final expression just just at this point you can substitute in the numerical values for V1 and V2 respectively you will get the answer for the what is this average speed for the entire tree okay so now you see a difference average speed is non zero average velocity is zero okay so that so that's the meaning of this question here. And when you log into the homework system, you will see this is one of the homework questions already. Hey, homework one. So, so thank you and hope you have a great day. See you again on Wednesday. Thank you.