Kinematic Formula for Displacement in the n-th Second

Kinematic Equations for Displacement in the $n$-th Second

In the study of kinematics, specifically regarding objects moving with constant acceleration, the transcript details a specific formula used to calculate the displacement of an object during a particular time interval. The expression provided is identified by the label Soth, which represents the displacement occurring within a specific $n$-th second of motion. In the context of the transcript, this is explicitly linked to the term "diablocement," a phonetic or transcribed variation of the term displacement. This calculation is a specialized case of the equations of motion and allows for the isolation of movement during a discrete one-second window, rather than calculating the total displacement from the commencement of motion.

Mathematical Formulation and Expression Components

The primary equation documented is $Soth = u + a(2n - 1)$. This formula consists of several critical physical variables that define the state of a moving body. The variable $u$ denotes the initial velocity of the object, which is the velocity at the start of the time measurement period ($t = 0$). The variable $a$ represents the constant acceleration experienced by the object throughout its duration of travel. The term $(2n - 1)$ relates to the specific $n$-th second for which the displacement is being calculated. For example, if one were interested in the displacement during the fifth second of motion, the value of $n$ would be $5$.

Technical Derivation and Physical Meaning

The physical significance of the formula $Soth = u + a(2n - 1)$ is rooted in the subtraction of two total displacement values. To find the distance traveled specifically between the end of the $(n - 1)$ second and the end of the $n$-th second, one typically subtracts the total displacement at the start of the interval from the total displacement at the end of the interval. Mathematically, this is expressed as follows:

Total displacement at time $n$: $S_n = u \times n + \frac{1}{2} \times a \times n^2$

Total displacement at time $n - 1$: $S_{n-1} = u(n - 1) + \frac{1}{2} \times a(n - 1)^2$

The displacement specifically in the $n$-th second is the difference: $s_n = S_n - S_{n-1}$. Upon simplification, this results in the standard kinematic identity, which the transcript approximates or records as $Soth = u + a(2n - 1)$. This specific form implies a relationship where the acceleration is applied to the time interval factor to determine the incremental change in position beyond the initial velocity component.

Identification of Transcription Labels and Identifiers

The transcript contains several unique identifiers and labels associated with the mathematical notes. The variable label "A" or "A_1" is positioned near the formula, possibly serving as a reference point for a specific gravitational constant or a specific problem instance labeled "A". Additionally, the numerical string "1449" is present, which may represent a reference code, a specific timestamp, or a numerical result relative to the displacement calculation. The term "ID" follows the main formula $u + a(2n - 1)$, which likely serves as an identification tag for the equation or a specific part of the document structure. Finally, "diablocement" is the specific terminology used by the source material to categorize the physical concept of displacement as it pertains to this derivation.