Points P,Q,R are in a vertical line such that PQ=QR.A. A ball at P is allowed to fall freely. What is the ratio of the descent through PQ and QR

 Let us assume

PR = 2x

So, PQ = QR = x

The ball begins with zero velocity.

Let us assume it takes time t1 to cover PQ and time t2 to cover QR.

Hence

S = UT + ½ AT2

=> 2x = 0 + ½ g(t1 + t2)^2

=> x = ¼ g(t1 + t2)^2

=> 4x = g(t1 + t2)^2 ……………..(i)

Also

x = 0 + ½ g(t1)^2

=> 2x = g(t1)^2 …………………(ii)

On dividing equation (i) by (ii),

(1)/(2) => 2 = (t1 + t2)2/(t1)2

=> [(t1 + t2)/t1]2 = 2

=> [t1+ t2/t1] = √2

=> t2/t1 = √2 – 1

=> t1 :t2 : 1 : (√2 – 1)