Quadratic drag equation by partial fractions

[matpo39]

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

 

[Dr Transport]

What is $$\int \frac{1}{1- \frac{v^{2}}{v_{0}}} dv$$, look it up in an integral table, you shouldn't have to resort to partial fractions, I suspect that it will be arctanh()...

 

[M98Ranger]

First of all, great job using partial fractions to solve the quadratic drag equation! You are on the right track.

Upon plugging in numbers, it is possible that you may have made a mistake in your integration or in plugging in the limits of integration. Double check your work and make sure that all of your calculations are correct.

Additionally, it is important to note that the natural log function is a complex equation and can sometimes give different results depending on how it is evaluated. It is possible that the slight discrepancy in your answers could be due to this.

Overall, your approach and use of partial fractions is correct. Keep working through the problem and double check your calculations to ensure accuracy. Good luck!