# Computing distance and time from a differential equation

1. Aug 5, 2011

hello,

having given 2 speeds v1 and v2 and the equation: k1*v'=k2-k3*v$^{2}$ how do i compute the distance traveled from v1 reaching v2 and the time needed. (k1,k2,k3 are constants).

i think i have to integrate to find the distance, but i just can't figure quite how to do it.

any tips are greatly appreciated.

2. Aug 5, 2011

### Pi-Bond

You need to find a function for velocity v(t). You can integrate v(t) to find the distance. To obtain this function, you can solve your differential equation using separation of variables:

$k_{1}\frac{dv}{dt}=k_{2}-k_{3}v^{2} \implies k_{1}\frac{dv}{k_{2}-k_{3}v^{2}} = dt$

3. Aug 5, 2011

### HallsofIvy

Staff Emeritus
And you can use "partial fractions" to integrate that: $k_1- k_2v^2= \left(\sqrt{k_1}- \sqrt{k_2}v\right)\left(\sqrt{k_1}+ \sqrt{k_2}\right)$

4. Aug 5, 2011