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CBSE Class 10 Answered

A,B,C and D are points in a vertical line such that AB =BC=CD. If a body falls freely through A, prove that the time of descent through AB,BC and CD are in ratio 1:root2-1:root3 - root2.      Give ans quickly in proper steps  
Asked by acv27joy | 12 May, 2018, 07:39: PM
answered-by-expert Expert Answer
Let AB=BC=CD= h ; Let t1 be the time taken to travel the distance AB.
speed u1 at end of the time t1, when the body is at B is given by
 
u1 = g×t1 .......................(1)
h = (1/2)×g×t1×t1................(2)
 
Let t2 be the time taken to travel the distance BC.
 
h = u1×t2+(1/2)g×t2×t2 = g×t1×t2 + (1/2)×g×t2×t2 ................(3)
 
initial speed during the travel of BC is substituted using (1) in the above relation.
Using the equations (2) and (3) and after canceling the common factors, we get
 
t12 = 2×t1×t2 + t22 ................(4)
 
By solving the quadratic equation for t2, we get t2 = t1×(√2-1)..................(5)
 
Speed u2 after travelling(t1+t2) seconds, when the body is at C is given by
 
u2 = g×(t1+t2) = g×t1×√2.............(6)
 
Let t3 be the time taken to travel the distance CD.
 
h = u2×t3+(1/2)g×t3×t3 = g×t1×√2×t3 + (1/2)×g×t3×t3 ................(7)
 
initial speed during the travel of CD is substituted using (6) in the above relation.
Using the equations (2) and (7) and after canceling the common factors, we get

t12 = 2×√2×t1 + t32 ................(8)
 
By solving the quadratic equation for t3, we get t3 = t1×(√3-√2)..................(9)
Answered by Thiyagarajan K | 13 May, 2018, 08:05: AM

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