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Derivation of the Kinematics Equations for Uniformly Accelerated Motion This derivation is based on the properties of a velocity-time graph for uniformly accelerated motion where the slope of the graph represents the acceleration graph's area represents the displacement Equation #1: slope = acceleration Starting with the slope where gives us our first equation: In this equation a represents the object's uniform acceleration t represents the time interval over which the object's velocity changed vf represents the object's final velocity at the end of the time interval vo represents the object's initial velocity at the beginning of the time interval Equation #2: rearrange equation #1 for vf Equation #3: area = displacement Equation #4: multiply equation #1 by equation #3 Equation #1: Equation #3: Equation #5: substitute equation #2 into equation #3 Equation #2: Equation #3: