### Modelling with Differentiation for A Level Maths and Physics

Motion under gravity - modelling with differentiation The motion of an object under gravity can easily be modelled by differentiation. Suppose the motion of a stone is given by the formula, y = 16t - 4t 2 , where y and t are displacement and time respectively. y = 16t - 4t 2 dy/dt = 16 - 8t When the stone reaches the highest point, dy/dt = 0 16 - 8t = 0 8t = 16 t = 2s Maximum height reached by the stone = 16x2 - 4x2 2 = 16m d 2 y/dx 2 = -8 < 0 Therefore, when t = 2s, the height reaches the maximum. In order to play with the simulation interactively, please use the following and play with the slider: