Computing distance and time from a differential equation

[vladgrigore]

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.

 

[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:

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

 

[HallsofIvy]

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

 

[vladgrigore]

thanks for that, i managed to figure out the integral but sadly it does not converge given the constants that i have.
thanks again

 

[PJC01]

I would suggest approaching this problem by first understanding the meaning and purpose of the given differential equation. This equation represents the relationship between velocity (v) and time (t) for an object moving at two different speeds (v1 and v2). The constants k1, k2, and k3 represent factors that affect this relationship.

To compute the distance traveled, we need to integrate the velocity function over the given time period. This will give us the total displacement of the object. However, since the equation involves a differential term (v'), you will need to use techniques such as separation of variables or substitution to solve the integral. It may also be helpful to plot the velocity function to visualize the behavior of the object over time.

As for computing the time needed, you can use the same approach by integrating the inverse of the velocity function. This will give you the time taken for the object to travel the desired distance. Again, use appropriate techniques to solve the integral and plot the function to gain a better understanding of the time behavior.

Overall, it is important to carefully analyze the given equation and use appropriate mathematical techniques to solve for distance and time. Additionally, it may be helpful to consult with a mathematician or use computational software to accurately compute these values.