Modeling ODE

Calculus Level 1

Suppose that we drop a stone, which falls freely with no air resistance. Experiments show that, under that assumption of negligible air resistance, the acceleration y = d 2 y d 2 t {y}^{\prime\prime}=\frac { {d}^{2}y }{ {d}^{2}t } of this motion is constant, i.e. equal to the so-called acceleration of gravity g = 9.8 m/sec 2 . g=9.8 \text{ m/sec}^2. State this as an ordinary differential equation for y ( t ) , y(t), the distance fallen as a function of time t . t.

y = g t {y}^{\prime\prime}=gt y = 1 2 g t 2 {y}^{\prime\prime}=\frac{1}{2}g{t}^{2} y = g {y}^{\prime\prime}=g y = g t + 1 2 g t 2 {y}^{\prime\prime}=gt+\frac{1}{2}g{t}^{2}

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