Ax_xiom
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- TL;DR Summary
- I am getting confusing results whenever I use differential equations to model the vertical motion of a dust particle
So I was experimenting with using differential equations to model motion and I wanted to use one to model the motion of dust being projected vertically upwards, and the differential equation I got was this $$\frac {dv} {dt} = -g-cv^2$$ where g is the acceleration due to gravity and c is a constant formed from air density, mass of the particle, drag coefficient and surface area which turns out to be about 0.787
When I solve the equation (with the boundary condition that the velocity at 0 is some set intial velocity) I get this: $$v = \frac {\sqrt{g} \tan\{\tan^{-1}\{\frac {\sqrt{c}} {\sqrt{g}} V_i\} - \sqrt{cg} t\}} {\sqrt{c}}$$
But when I plot and intergrate this I get weird results. If I set the inital velocity to the speed of light the model predicts it would only travel 23m upwards which makes no sense. So what happened and what did I do wrong here?
When I solve the equation (with the boundary condition that the velocity at 0 is some set intial velocity) I get this: $$v = \frac {\sqrt{g} \tan\{\tan^{-1}\{\frac {\sqrt{c}} {\sqrt{g}} V_i\} - \sqrt{cg} t\}} {\sqrt{c}}$$
But when I plot and intergrate this I get weird results. If I set the inital velocity to the speed of light the model predicts it would only travel 23m upwards which makes no sense. So what happened and what did I do wrong here?