Understanding the Distance Covered by a Falling Body: nth Second Analysis

When studying the movement of a body falling to the Earth, it is essential to understand the distances it covers in each second. The distances covered each second form an interesting mathematical series. In this article, we will delve into the details of this series, its properties, and how to calculate the distance covered in the nth second.

Series Analysis

Consider the sequence of distances covered by a body falling to the Earth during each second. The distances covered are as follows:

1st second: 3a feet 2nd second: 5a feet 3rd second: 7a feet nth second: 2an - a feet

From the above, we can observe that the distances covered each second form an Arithmetic Progression (AP) with the first term being 3a and the common difference being 2a.

Formulas and Derivations

The nth term of an AP can be represented as:

Tn a (n - 1) x common difference

For our sequence, the common difference is 2a:

Tn 3a (n - 1) x 2a

Which simplifies to:

Tn 3a 2an - 2a 2an - a

Further simplification gives:

Tn a(2n - 1)

Summation of the Series

The Tn terms of this AP can be summed to express the total distance covered in n seconds. The formula for the sum of an AP is:

Sum Sn n/2 [2a (n - 1) x 2a]

Substituting the given values:

Sum n/2 [2a 2an - 2a] n/2 [2an] an^2

Conclusion

Understanding the series and the formula for the nth term is crucial for a comprehensive analysis of the falling body's motion. This approach helps in identifying the exact distance covered in any given second, making it a valuable tool in physics and engineering.

By studying these sequences and formulas, we can better understand the dynamics of falling objects and apply this knowledge in various practical scenarios.