 Welcome, friends, to this new session on problem solving. And now the question is, a point traversed half the distance with velocity v0. It's given. The remaining part of the distance was covered with velocity v1, so that's for half the time. That is, in the remaining journey, half the time, the velocity was v1, and v2 for the other half of the time, you have to find mean velocity. Mean velocity or the average velocity of the point over the whole time of motion. Now, I have drawn a representative diagram over here, if you can see. So a2c is the distance traveled. Now, b is exactly half the distance, so hence, ab is l by 2. And bc also is l by 2, if you can see here. Now, I have also mentioned some important information. So for example, in the first half of the journey, v0 is the speed, and t1 is the time taken. In the second part of the journey, let's say x distance is traveled with v1 velocity, and time taken is, let's say, t. And the other half of, or the remaining part of the journey is l2 minus x. This is the distance, and the velocity was v2. And time taken was same as the previous half, that is t. Now, so total distance traveled is l, we all know. We are assuming it to be l. Now, ab is l by 2, distance traveled with velocity v0. So t1 is nothing but distance upon speed. So t1 is l by 2 divided by v0. Now, let us see bd. What is bd? So this distance basically is bd. So x upon t will be v1. x upon t is v1, because distance by time. So hence, t will be x upon v1 by manipulation. Now, similarly, dc, if you see, distance traveled is l2 minus x. Divided by time is v2. Now, mind you, all these are uniform velocity cases. So hence, we are able to write these equations. Now, e is equal to l by 2 minus x divided by v2. So we found t. Now, you can equate 2 and 3. But before that, what is this mean velocity is given or mean speed is given by? Since it is a straight line motion, so mean speed will be equal to mean velocity. So l is t1 plus t plus t. So total distance was l. And total time taken was t1, then t plus t. So this is an average velocity. So from 2 and 3, as we discussed earlier, we equate these two equations. So you will get this relationship. So x by v1 equals l by 2 minus x by l by 2 minus x upon v2. So hence, if you simplify it, you will get x equals this v1 upon v1l upon 2 v1 plus v2. So hence, t is nothing but x by v1 from 2. We know that t is equal to x by v1 here. So hence, t is equal to x by v1, which is v1. So now you can put the value of x in this equation. So it becomes v1l upon 2 times v1 plus v2. And then divide by v1 upon simplification, you will get t is equal to l upon 2 times v1 plus v2. Now mean velocity is simple. v mean velocity is nothing but total distance l divided by total time. So l upon t1 plus 2t. So l and t1, I know, we just found out t1 is equal to, from here, what is t1? t1 is this from 1, t1. And then t, we just found out here. So if you substitute it and simplify, you will get this final relationship. So hence, you will get finally, if you substitute all these t1 and t into this relationship, you will get mean velocity as 2v0 v1 plus v2 divided by 2v0 plus v1 plus v2.