Newton’s law is fundamental to our work in mechanics. Newton’s law relates the applied force, F(t), to the resulting acceleration, a(t), by F(t) = m a(t) (1) Here the mass, m, is assumed to be constant and t is the time. Equation (1) is differential equation that predicts where the mass m will be at any time if F(t) is a known, specified function and the appropriate initial conditions are given. To realize this, the acceleration may be written as the second derivative of time, or d2x a(t) 2 (2) dt where d denotes the total derivative and x denotes the displacement. Substitution of Equation (2) into Equation (1) yields: d 2 x(t) F(t) (3) dt 2 Equation (3) shows explicitly that Newton’s law results in a second order differential equation.
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