Deriving and Integrating Terminal Velocity Equation for Position

[Joza]

(dv_y)/(v_y+v_t) = (k/m)(dt) where v_y is y velocity, v_t is terminal velocity, k is fluid resistance constant, m is mass and t is time

This is my equation I derived for terminal velocity. I am going to integrate it for a function for position.

In my text, the limits of integration on the left side are written as V and 0. Right side is t and 0.

Could someone clear up the limits here? Like, there are 2 vs, v_y and v_t..what v is the limit referring to here? And I am a bit confused as to why the limits are needed here? I understand integration, but I am fairly new to it, maybe that's why!

 

[Hootenanny]

Integrating the differential you have there won't give to position as a function of time, it will give you velocity as a function of time.

 

[ogb p]

First of all, great job on deriving an equation for terminal velocity! It shows that you have a good understanding of the concept and are able to apply it in a mathematical way. Now, let's talk about integrating it for a function of position.

The limits of integration in this case are referring to the initial and final values of the variable being integrated. In this case, the left side of the equation is being integrated with respect to time (t), so the limits are 0 and t. This means that we are finding the area under the curve between the initial time (0) and the final time (t).

On the right side of the equation, we have a constant (k/m) multiplied by dt, which is essentially just a small change in time. This represents the change in velocity over a small period of time, and by integrating it, we are finding the total change in velocity over the entire time period.

As for the confusion with the two "v"s, the limit on the left side (v and 0) is referring to the variable v_y, which is the y velocity. This means that we are finding the area under the curve of the y velocity with respect to time. On the right side, we have v_t, which is the terminal velocity. This is a constant value, so it does not need to be integrated.

I hope this helps to clarify the limits and the overall process of integrating for a function of position. Keep up the good work!