Question

A body is falling freely under gravity from a certain height from the ground. If the sum of the displacements of the body in the second and third seconds of its motion is 32% of the height from which it is falling, then the speed with which the body hits the ground is (Acceleration due to gravity = $10~\text{m s}^{-2}$)

• A. $25~\text{m s}^{-1}$
• B. $50~\text{m s}^{-1}$
• C. $100~\text{m s}^{-1}$
• D. $75~\text{m s}^{-1}$

Hint:

Use the formula for displacement in nth second: \(s_n = u + \frac{1}{2}g(2n-1)\). Sum of displacements over specific seconds can be used to calculate total height and final velocity.

Answer 1

The Correct Option is B

Let initial velocity \(u = 0\), acceleration \(g = 10~ms^{-2}\). Displacement in nth second: \(s_n = u + \frac{1}{2}g(2n-1) = 5(2n-1)\). Sum of 2nd and 3rd second displacements: \(s_2 + s_3 = 5(3) + 5(5) = 15 + 25 = 40\) units. This is 32% of height, so total height \(h = 125\). Velocity at impact \(v = \sqrt{2gh} = \sqrt{2 \cdot 10 \cdot 125} = \sqrt{2500} = 50~ms^{-1}\).