 Hey hello friends welcome again to another session on problem solving and we are taking up problems from mechanics chapter in physics and this is related to motion and we are talking about uniform and you know uniform speed and velocity okay. So this question says that a car travels first half time first half time with the uniform speed again this is the case of uniform speed and that is u for the first half time he travels at a uniform speed of u and the next half time with a uniform speed of v and the demand of the question is find the average speed okay. So in the previous session if you noticed the case was the object was travelling for half the distance with the constant speed and next half distance with the constant speed v in this case it's they are talking about time right half the time so it's speed u and next half the time speed v. So as discussed in the previous case first step should be draw a representative diagram always helps in understanding the problem. So let's say this is a total distance travel and it's always good practice to you know label it let's say a is the initial point or position b is the final position and let's say it is in the first half time it covers a distance d1 let's say now what one general misconception is in half time they will travel the half the distance no that's not correct why because the speeds are different for the different timings. So hence don't say that they will be covering the same distance same amount of distance in both the half times because so let's say a journey of 100 second was there and the first 50 second the speed was different 20 meters per second and let's say in the second 50 second half of the journey that is from 51st second to 50 you know from 50 second onwards 200 seconds and in the second half of the journey that is from 50 second onwards 200 seconds let's say the speed was 40 meters per second so the speeds are different so obviously in the same time they would be traveling different amount of distances I hope that is clear to all of you now it's also given that the velocity or the speed rather you know is u in the first case and from p to b it is v and it's also given that throughout the journey it's constant so this is constant here constant speed constant speed journey or that's called uniform motion you know this and this is a constant speed constant speed again right here it is u here it is v now these are straight line in a straight line they are traveling their uniform motion otherwise speed anyways is going to be constant in uniform motion velocity is constant where direction as well as speed has to be constant so please have the clarity in your mind now in this problem they are not talking about velocities at all they are talking about speeds we need not be a straight line but anyway so half the distance traveled or not half the distance let's say d1 distance traveled in half the time so let's say this total time taken for the entire journey let's say that is capital T okay this is a total time taken so all the variables have been mentioned in the diagram itself self-representative right so hence this is t upon 2 I hope this is clear because this is half time and this one also this journey took t by 2 time you know seconds right okay now let's try to go for the you know what is the conceptual understanding here so we know that speed is equal to distance upon time distance traveled upon time taken for that distance travel okay so there are two cases what is this case case one is from a 2 point p what is the distance d1 what is the time taken capital T by 2 and what's the speed u so how are they related u is equal to d by t upon 2 right distance divided by sorry d1 not d d1 right so what is d1 guys if you see d1 will can I write d1 as d1 will be u into t upon 2 right cross multiplication simple mathematics now case 2 what is case 2 when the body is or the car is moving from p2 be clear what is the speed in this case v what is the distance traveled d2 and what is the time taken capital T by 2 all mentioned in the diagram check right so what can I say v is equal to d2 upon t by 2 again is it it so can I say d2 is equal to v into t by 2 right this is d2 yeah this is d2 very good so let us you know it's all also a good practice to keep numbering the equations or results which you get so 1 and 2 like that okay now the demand of the question let's go to that so average speed average average speed will be nothing but what will be the average speed for any case it's total distance traveled total distance traveled divided by total time taken isn't it total time taken so let's find out the total distance total distance was a b and how much was that d1 plus d2 is the total distance some of the two distances divided by capital T total time taken was capital T isn't it let's see if you can eliminate all these variables what is d1 if you check equation number one it is u into capital T by 2 and what is d2 check equation number two it's v into t by 2 and divided by capital T I hope this exercise is clear now what happens you can take t by 2 as common and what is there u plus v in the numerator and in the denominator is it is t so hence if you cancel this t and this t you will get u plus v by 2 and dear friend this is the average velocity so the time span is same the average velocity is nothing but the average of the other mean of the two velocities right so hence if it goes in the same time from one point to the other with you and from this point to the other point with v if the time taken is same the average velocity for the entire journey will be u plus v by 2 right this is the learning you can actually keep this result in your mind and let's say whenever you are facing a mcq type questions in any comparative exams like ntsc kvy or rather for that matter engineering entrances or neat entrances as well you can keep this result in mind