A Discrete-Time Summation Derivation of the Second Equation of Motion
- 1Independent Researcher
Abstract
The second equation of motion, s = ut + ½at², is usually derived by integrating velocity or from the area under a velocity-time graph. This paper presents an alternative, calculus-free derivation based on summing the displacement covered in successive unit-time intervals. Assuming constant acceleration and using the velocity at the start of each interval, the total displacement is found to be s = ut + ½at² − ½at. The extra term −½at is shown to be a discretisation error caused by ignoring the change of velocity within each interval. Generalising to a time step h gives s = ut + ½at² − ½ath, which reduces to the standard equation as h → 0. Using the mid-interval velocity removes the error exactly. The method gives students an elementary, summation-based explanation of where the factor ½ in ½at² comes from and links kinematics to the idea of Riemann sums.
Independent student research. This preprint has not been peer-reviewed. An AI assistant was used to check the algebra and help with writing.
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BibTeX
@misc{asthana2026discretetime,
title = {{A Discrete-Time Summation Derivation of the Second Equation of Motion}},
author = {Amritanshu Asthana},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23657193},
url = {https://zenodo.org/records/23657193},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Discrete-Time Summation Derivation of the Second Equation of Motion
AU - Amritanshu Asthana
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23657193
UR - https://zenodo.org/records/23657193
AB - The second equation of motion, s = ut + ½at², is usually derived by integrating velocity or from the area under a velocity-time graph. This paper presents an alternative, calculus-free derivation based on summing the displacement covered in successive unit-time intervals. Assuming constant acceleration and using the velocity at the start of each interval, the total displacement is found to be s = ut + ½at² − ½at. The extra term −½at is shown to be a discretisation error caused by ignoring the change of velocity within each interval. Generalising to a time step h gives s = ut + ½at² − ½ath, which reduces to the standard equation as h → 0. Using the mid-interval velocity removes the error exactly. The method gives students an elementary, summation-based explanation of where the factor ½ in ½at² comes from and links kinematics to the idea of Riemann sums. Independent student research. This preprint has not been peer-reviewed. An AI assistant was used to check the algebra and help with writing.
ER -