 hello friends so here is another question it says derive mathematically the first equation of motion v is equal to u plus at here this v should have been with small letter so never mind v is equal to u plus at read the question carefully it says derive mathematically right so hence we will have to use mathematical tools to derive this equation okay now what is v stand for what does we stand for so v is final velocity this already we know final velocity so you should write like this final velocity so when it is said final that means there must be some initial also so that is u so you know u is initial velocity is it now what is this initial and final business so basically if you have let's say a journey happening in a straight line again this is only true for journey happening in a straight line always remember okay or you are treating v and u as speeds right then even in curved surface it is fine but as far as ninth grade portions are concerned we are going to restrict our analytics to straight line motion let's say this is the starting point your home this is your final endpoint let's say your school and though your this is not a practical case why because there is no straight distance between a and straight straight line motion between a and b you take a lot of you know you have to travel like this through the city and you reach here but let's say we draw a straight line and let's assume that you are flying an aircraft from a to b horizontally okay so here when you started your you know your speed is you now you can say how can someone start not from zero but from something else now you know here is the thing so hence basically this analytics is something like this that you are going in a long path and then i am concentrating on these this window window of when you were at t equals to t1 to let's say t is equal to t2 the motion was happening from any any other point let's say from here t and you are going from t to q and i am analyzing your motion between these two times or between the time interval t1 to t2 this point is a as mentioned above and this point is b okay so when there was time t1 at your clock by your clock so you were at point b and at that moment you were traveling in this direction with a velocity u and when the your clock had was showing a time t2 let's say t1 is 12 p.m t2 is 12 30 p.m that t2 that is at 12 30 p.m you are moving with the velocity v right so your velocity has changed as you were traveling from a to b and what is the characteristic of this motion so you were traveling with an acceleration a in this direction okay so you are basically accelerating what does that mean that means your speed in this direction or your velocity is going to increase as you travel from a to b and finally it has reached a value v when you are at you know location b at time t is equal to t2 okay now by the definition of acceleration guys what is acceleration so acceleration a is defined as change in change in velocity change in velocity divided by change in time change in time so what all velocities and what all time so hence if you are considering this window from t1 to t2 i'm calling it a window it's a time interval right at t equals to t1 you are at a at time t equals to t2 you are at b so hence you are traveling from a to b and the changes you are seeing is that at a where you are traveling with a velocity u and at b you're traveling at a velocity v and this change in velocity was because you had an acceleration attached with you okay so this is where this is what the scenario is so hence the definition a can be rewritten as a is equal to v2 minus v1 divided by t2 minus t1 isn't it we can always write that change in velocity is v2 minus v1 and change in time is t2 minus t1 now the assumption here what we are making is the acceleration is constant acceleration is constant what does that mean it means that let's say you started with zero meter per second okay and at the end of one second your speed is let me draw a table okay so here is t here is t and here is your velocity so at the at zero you are having zero meter per second speed at the end of one second you have one meter per second speed at the end of second second that is at the end of two seconds your speed is two meter per second so you're seeing your you know your velocity is increasing at the end of third second your speed is three meter per second right so as if you know the velocity is increasing constantly right so as if if you throw something down it starts increasing the speed starts increasing isn't it but similar to that situation this is there so if you see for every gap of one second every gap of one second your speed goes up by how much two minus one one meter per second isn't it so every one second you add one meter per second to your speed I'm using speed and velocity interchangeably here why because we are concentrating only on straight line motion where speed is equal to velocity if the direction is same correct so straight line motion same direction speed is as good as velocity value wise okay so what I'm trying to explain here is after every one second so in the time gap of one second your speed goes up by one meter per second so you can see this one and this one is constant so after let's say fourth second you can predict now your speed will be four meter per second so again you added one meter per second to your speed this is the case of constant acceleration constant acceleration now let's say the same thing would have happened that after end of one second every one second interval your speeds are increasing so let's say I draw another table here now time stays the same but your speeds are like this zero one then four then nine like that your speeds are increasing so at the gap of or at the end of every one second right for every interval of one second your speed right what is the change in speed so here you see one is the change what is the change here three is the change what is the change here five is the change so so your change is not constant is it change in velocity is not constant it is varying with every second and hence we this is the case of non uniform acceleration or yes or yeah non uniform acceleration so please understand these two scenarios very very clearly this is non-uniform acceleration here uniform acceleration okay so once you have understood so in your grade we are restricting only to uniform acceleration that is the acceleration is not changing with time acceleration is constant this value is constant at you know every two time intervals you take you find the change in velocity find the change in divide the change in velocity by the time taken in that change you will get same value of k now from here it is simply mathematical so a times t2 minus t1 is equal to v2 minus v1 is it it so we can say v2 is equal to v1 plus a times t2 minus t1 and what is v1 and v2 so in this case you can clearly see v1 is nothing but u initial velocity v2 is nothing but v final velocity right t1 I am considering to be zero start-off motion let's say your your clock starts at t1 equals to zero and t2 is equal to let's say t if you deploy all these values in the given equation what will you get you'll get v is equal to u plus a t and this is what is the mathematical derivation of this particular equation of motion so always remember v is equal to u plus a t is derived from the definition of acceleration itself okay I hope you understood this problem