 Hello and welcome to the session. Let's work out the following problem. It says p q r s r points in a vertical line so that p is the highest and p q is equal to q r is equal to r s if Body falls from rest at p prove that the times of describing the successive intervals are in the ratio 1 is to root 2 minus 1 is to root 3 minus root 2. So let's now move on to the solution We are given that p q r s r points the vertical line so that p q is equal to q r is equal to r s Let this be equal to h As this much as h this much as h and this much as s h so P s is equal to 3 h Now we have to find the time taken to descend the distance p q q r and r s now p q is h q r is p r minus p q that is 2 h minus h and similarly r s is equal to p s minus p q plus q r that is 3 h minus 2 h right now let p 1 p 2 and t 3 be the time taken to descend the distance p q q r and R s respectively since the motion is vertically downwards so h is equal to 1 by 2 gt square right and Since here the time taken to this descent distance p q is t 1 so H will be 1 by 2 into g t 1 square so this implies t 1 is equal to under the root 2 h by g now for the let Capital t be the time taken to descend the distance P r which is 2 h so 2 h is equal to 1 by 2 g t square since capital t is the time taken to descend the distance P r That is 2 h so this implies Capital t is equal to under the root 4 h upon g Now we have assumed that t 2 is the time taken to descend the distance q r so small t 2 is equal to time taken to descend p r minus time taken to descend q now time taken to descend p r is under the root 4 h by g and Time taken to descend p q is t 1 which is under the root 2 h by g now let p be the time taken to descend s Which is 3 h therefore 3 h is equal to 1 by 2 g t square So this implies p is equal to under the root 6 h upon g Now we have assumed that Small t 3 is the time taken to descend the distance r s Which is equal to time taken to descend p s minus Time taken to descend p r now time taken to descend p s is under the root 6 h upon g and Time taken to descend p r is 4 h upon g under the root 4 h upon g right now we have to prove that the times of Describing the successive intervals are in the ratio 1 is to root 2 minus 1 is to root 3 minus root 2 That means we have to find the ratio of t 1 t 2 and t 3 T 1 is to t 2 is to T 3 now t 1 is under the root 2 h upon g t 2 is Under the root 4 h upon g minus under the root 2 h upon g and t 3 is under the root 6 h upon G minus under the root 4 h upon g cancelling h by g from all the ratios we have Under the root 2 is to under root 4 minus under root 2 is to Under root 6 minus under root 4 this is again under root 4 My is to under the root 2 into 2 minus under the root 2 is to under the root 2 into 3 Minus under the root 2 into 2 Now cancelling root 2 we have 1 is to root 2 minus 1 Is to root 3 Minus root 2 and this is what we have to prove the ratio is 1 is to under the root 2 minus 1 is to under the root 3 minus under the root 2 So this completes the question and the session bye for now take care. Have a good day