m d v y d t = − m g − k v y m d v y m g + k v y = − d t ∫ v 0 s i n θ v y m d v y m g + k v y = − ∫ 0 t d t m k ln ( v y + m g k ) | v 0 s i n θ v y = − t ln ( v y + m g k ) − ln ( v 0 s i n θ + m g k ) = − k t m ln ( v y + m g k v 0 s i n θ + m g k ) = − k t m v y + m g k v 0 s i n θ + m g k = e − k t m v y = ( v 0 s i n θ + m g k ) e − k t m − m g k d y d t = ( v 0 s i n θ + m g k ) e − k t m − m g k d y = [ ( v 0 s i n θ + m g k ) e − k t m − m g k ] d t ∫ 0 y d y = ∫ 0 t [ ( v 0 s i n θ + m g k ) e − k t m − m g k ] d t y = m k ( v 0 s i n θ + m g k ) ( 1 − e − k t m ) − m g t k {\displaystyle {\begin{aligned}m{\frac {dv_{y}}{dt}}&=-mg-kv_{y}\\{\frac {m\,dv_{y}}{mg+kv_{y}}}&=-dt\\\int _{v_{0}sin\theta }^{v_{y}}{\frac {m\,dv_{y}}{mg+kv_{y}}}&=-\int _{0}^{t}\,dt\\\left.{\frac {m}{k}}\ln(v_{y}+{\frac {mg}{k}})\right\vert _{v_{0}sin\theta }^{v_{y}}&=-t\\\ln(v_{y}+{\frac {mg}{k}})-\ln(v_{0}sin\theta +{\frac {mg}{k}})&=-{\frac {kt}{m}}\\\ln({\frac {v_{y}+{\frac {mg}{k}}}{v_{0}sin\theta +{\frac {mg}{k}}}})&=-{\frac {kt}{m}}\\{\frac {v_{y}+{\frac {mg}{k}}}{v_{0}sin\theta +{\frac {mg}{k}}}}&=e^{-{\frac {kt}{m}}}\\v_{y}&=(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}\\{\frac {dy}{dt}}&=(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}\\dy&=[(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}]dt\\\int _{0}^{y}dy&=\int _{0}^{t}[(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}]dt\\y&={\frac {m}{k}}(v_{0}sin\theta +{\frac {mg}{k}})(1-e^{-{\frac {kt}{m}}})-{\frac {mgt}{k}}\\\end{aligned}}}
![{\displaystyle {\begin{aligned}m{\frac {dv_{y}}{dt}}&=-mg-kv_{y}\\{\frac {m\,dv_{y}}{mg+kv_{y}}}&=-dt\\\int _{v_{0}sin\theta }^{v_{y}}{\frac {m\,dv_{y}}{mg+kv_{y}}}&=-\int _{0}^{t}\,dt\\\left.{\frac {m}{k}}\ln(v_{y}+{\frac {mg}{k}})\right\vert _{v_{0}sin\theta }^{v_{y}}&=-t\\\ln(v_{y}+{\frac {mg}{k}})-\ln(v_{0}sin\theta +{\frac {mg}{k}})&=-{\frac {kt}{m}}\\\ln({\frac {v_{y}+{\frac {mg}{k}}}{v_{0}sin\theta +{\frac {mg}{k}}}})&=-{\frac {kt}{m}}\\{\frac {v_{y}+{\frac {mg}{k}}}{v_{0}sin\theta +{\frac {mg}{k}}}}&=e^{-{\frac {kt}{m}}}\\v_{y}&=(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}\\{\frac {dy}{dt}}&=(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}\\dy&=[(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}]dt\\\int _{0}^{y}dy&=\int _{0}^{t}[(v_{0}sin\theta +{\frac {mg}{k}})e^{-{\frac {kt}{m}}}-{\frac {mg}{k}}]dt\\y&={\frac {m}{k}}(v_{0}sin\theta +{\frac {mg}{k}})(1-e^{-{\frac {kt}{m}}})-{\frac {mgt}{k}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aecc0c420524a24ebed18d23ec07812a16458571)
