text: # 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