Mplwp ballistic trajectories velocities

${\displaystyle {\ddot {\vec {r}}}(t)=-k\,v\cdot {\vec {v}}-g\cdot {\hat {\vec {y}}}}$

${\displaystyle {\vec {r}}(0)={\vec {0}},\;{\dot {\vec {r}}}(0)=v_{0}({\hat {\vec {x}}}\cos(\alpha _{0})+{\hat {\vec {y}}}\sin(\alpha _{0}))}$

The parameters are:

• ${\displaystyle g=1}$, ${\displaystyle k=1}$, ${\displaystyle \alpha _{0}=45^{\circ }}$
• The initial velocity takes the values ${\displaystyle v_{0}=2}$, ${\displaystyle v_{0}=4}$, ${\displaystyle v_{0}=6}$, ${\displaystyle v_{0}=8}$, ${\displaystyle v_{0}=10}$