Question

If a stone dropped from the top of a tower travels half of the height of the tower during last second of its fall, the time of the fall is ( in seconds)
A. $13 + \sqrt 2$
B. $2.4 + \sqrt 2$
C. $3.2 - \sqrt 2$
D. $4.2 + \sqrt 2$

Answer 1

The above problem can be resolved using the concept and the fundamentals of the body undergoing free fall. The mathematical equation for the distance travelled by any object in a particular fraction of time and the total distance within the total time. Moreover, the particular equations are further resolved by applying the quadratic formula.

Complete step by step answer:
Let the height of the tower be H.
And the stone takes n seconds to reach the ground.
Then the distance covered in the ${n^{th}}$ second is,
${D_n} = \dfrac{1}{2} \times D$
Here, D denotes the distance covered in n seconds. And its value is,
$D = \dfrac{g}{2}{n^2}................................\left( 1 \right)$
Here, g is the gravitational acceleration.
The distance in the ${n^{th}}$ second is,
${D_n} = \dfrac{g}{2}\left( {2n - 1} \right).........................\left( 2 \right)$
Comparing the equation 1 and 2 as,

\dfrac{g}{2}\left( {2n - 1} \right) = \dfrac{g}{2}{n^2}\\\ {n^2} - 2n + 1 = 0 \end{array}$$ On solving the above the quadratic equation, the values obtained as, $$n = 2 \pm \sqrt 2 $$ This value is close to the value of $$2.4 + \sqrt 2 $$. Therefore, the time of fall is $$2.4 + \sqrt 2 $$ seconds **So, the correct answer is “Option B”.** **Note:** In order to resolve the given problem, the fundamental relation for the distance travelled in a specific time, and the total distance travelled for the whole journey needs to be remembered. Moreover, the free fall of the body is influenced by the gravitational force acting on the body. Moreover, the key concept of the factors affecting the motion of the body in the vertical direction also need to be considered to resolve the given problem.