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
explanation: The document is a discussion thread from a forum where a user asks how to compute the distance traveled and the time needed given two speeds and a differential equation. The user seeks tips on integrating the equation to find the distance. Two other users, Pi-Bond and HallsofIvy, provide guidance on solving the differential equation using separation of variables and partial fractions, respectively.
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
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distilabel_metadata.content: # 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread from a forum where a user asks how to compute the distance traveled and the time needed given two speeds and a differential equation. The user seeks tips on integrating the equation to find the distance. Two other users, Pi-Bond and HallsofIvy, provide guidance on solving the differential equation using separation of variables and partial fractions, respectively.
  
  Label: question-and-answer