- #1
matpo39
- 43
- 0
hi, i am trying to show that
dv/(1- (v^2/v_ter^2)) = g*dt which after integrating is
v=v_ter*tanh(g*t/v_ter) (motion with quadratic drag) can also be obtained by using natural logs.
so far i have this:
letting u = v/v_ter
i can use partial fractions to get
du/(1-u^2) = 1/2 *(1/(1+u) + 1/(1-u)) *du
then using my limits of integration as 0 to u , i get
1/2* [ln(1+u) + ln(1-u)] = g*dt
then integrating the other side i get as my final equation
1/2 *[ ln(1+v/v_ter) + ln(1-v/v_ter)] = g*t
but when i tried to plug numbers into each equation the numbers didnt match.
does anyone know what i may have done wrong?
thanks
dv/(1- (v^2/v_ter^2)) = g*dt which after integrating is
v=v_ter*tanh(g*t/v_ter) (motion with quadratic drag) can also be obtained by using natural logs.
so far i have this:
letting u = v/v_ter
i can use partial fractions to get
du/(1-u^2) = 1/2 *(1/(1+u) + 1/(1-u)) *du
then using my limits of integration as 0 to u , i get
1/2* [ln(1+u) + ln(1-u)] = g*dt
then integrating the other side i get as my final equation
1/2 *[ ln(1+v/v_ter) + ln(1-v/v_ter)] = g*t
but when i tried to plug numbers into each equation the numbers didnt match.
does anyone know what i may have done wrong?
thanks