 Hello and welcome to the session let's work out the following problem. It says a stone is projected at an angle alpha with the horizontal. Given that its velocity when it attains half of the maximum height it can attain is root 2 times of the velocity at the maximum height prove that tan alpha is equal to root 2. So let's now move on to the solution we are given that a stone is projected making an alpha with the horizontal. So let the stone be projected velocity u initial velocity u at an angle alpha ab be the maximum height pq half the maximum height therefore ab is 2 square sin square alpha upon 2g and since pq is half the ab so it is u square sin square alpha upon 4g. Small v is the velocity then v square is given by the formula u square minus 2g h. Now h is the height at p which is u square sin square alpha upon 4g so it is 2g into pq which is u square minus 2g into u square sin square alpha upon 4g. So now this is equal to u square minus 2g gets cancelled with 2g here and we have u square sin square alpha upon be the velocity u cos alpha that is a horizontal component of the velocity is u cos alpha at a so we are given that velocity when the stone attains half of the maximum height is root 2 times of the velocity at the maximum height here we have v1 and the velocity when the stone attains half of the maximum height it is root 2 times of the velocity when it is at the maximum height so v is equal to root 2 times u cos alpha. So this implies v square is equal to square cos square alpha now we have v square is equal to u square minus u square sin square alpha by 2 let us name this as 1 and this is 2 so equating 1 and 2 we have u square minus u square sin square alpha upon 2 is equal to 2u square cos square alpha this is from 1 and 2 now this implies u square into 1 minus sin square alpha by 2 is equal to 2u square cos square alpha u square gets cancelled on both sides so we have 2 minus sin square alpha is equal to 4 cos square alpha taking LCM simplifying we have this and this implies 2 is equal to 4 cos square alpha plus sin square alpha and this implies 2 is equal to 4 3 cos square alpha plus cos square alpha 4 cos square alpha can be written as 3 cos square alpha plus cos square alpha plus sin square alpha. So this implies 2 is equal to 3 cos square alpha plus 1 as we know that cos square alpha plus sin square alpha is 1. So this implies 1 is equal to 3 cos square alpha subtracting 1 from 2 we have this. So this implies cos square alpha is equal to 1 by 3 and this implies cos alpha is equal to 1 by root 3. Now cos alpha is given by base upon hypotenuse. So in the right triangle base is 1 and hypotenuse is root 3. Now we can easily find out the perpendicular. So the perpendicular is equal to under the root of hypotenuse square minus the base square. So this is equal to under the root 3 root 3 square is 3 minus 1 and this is equal to root 2. So the perpendicular is root 2. Now the sin alpha is given by perpendicular upon hypotenuse. Now perpendicular is root 2 hypotenuse is root 3. Now we have obtained sin alpha and cos alpha so tan alpha is equal to sin alpha upon cos alpha sin alpha is root 2 by root 3 cos alpha is 1 by root 3. So this is equal to root 2 by 1 which is equal to root 2 and this is what we had to prove. So this completes the question and the session. Bye for now. Take care and have a good day.