Question

A block of mass m is dropped from the fourth floor of an office building and hits the sidewalk below at speed v. From what floor should the block be dropped to double that impact speed?
(A) the sixth floor
(B) the eighth floor
(C) the tenth floor
(D) the twelfth floor
(E) the sixteenth floor

Answer 1

Hint
A conservative force only exists when the work done by the force that acts on an object is free of the object's path. Alternatively, the work done by a conservative force depends only on the end points of the motion. An example of a conservative force is gravity.

Complete step by step answer
As the block of mass $m$ that falls freely on the sidewalk from the fourth floor of the building, only the conservative gravitational force acts on the block of mass $m$. That is, the mechanical energy that acts on the block of the mass $m$ is conserved or preserved.
$\Rightarrow K{E_f} = P{E_f} = K{E_i} + P{E_i}$
Where, $K{E_f}$ denotes the final kinetic energy of the block, $P{E_f}$ denotes the final potential energy of the block, $K{E_i}$ denotes the initial kinetic energy of the block, $\Rightarrow K{E_i}$ denotes the initial kinetic energy of the block.
Let’s assume that the block is released from rest that is the initial kinetic energy is equal to zero, and by taking $y = 0$ at ground the level the final potential energy is equal to zero, that is
$\Rightarrow K{E_f} = P{E_i}$ or $\dfrac{1}{2}mv_f^2 = mgy$
Where, ${v_f}$ denotes the final velocity of the block, $g$ denotes the acceleration due to gravity. ​​
$\Rightarrow {y_i} = \dfrac{{v_f^2}}{{2g}}$
That is, to double the final speed of the block, it is essential to increase the initial height by the factor of four.
Therefore, the block should be dropped to double that impact speed is the sixteenth floor.
Hence, the option (E) the sixteenth floor is the correct answer.

Note
We can deduce that the gravitational force does not depend on the path taken but only depends on the initial position and final position of the object. Hence, the gravitational force is a form of conservative force.