A Discrete-Time Summation Derivation of the Second Equation of Motion

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 This preprint has not been peer-reviewed.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.