Question

I don't understand the following equation change in classical mechanics...

Answer 1

\[\frac{ m }{ k } \int\limits_{0}^{v_0} \frac{ vdv }{ v + \frac{ m }{ k }g }\]

Answer 2

to

Answer 3

\[\frac{ m }{ k } \int\limits_{0}^{v_0} 1 - \frac{ \frac{ m }{ k }g }{ v + \frac{ m }{ k } g}dv\]

Answer 4

I just dont see how the equation is changed in this way, but it's the instructors solution and makes the integral doable during a test format. Any help?

Answer 5

In the first equation add and subtract the following term in the integrand\[\frac{ (v +\frac{ m }{ k }g) }{ (v +\frac{ m }{ k }g) } \]
then collect terms to get the second equation

Answer 6

This is a common technique in changing the form of an expression without changing its value. ie adding and subtracting an expression at the same time.

Answer 7

Im used to multiplying an exp by an exp equal to one, but adding and subtracting is a little less common, or was, during my school so far. Ill try to work through it and see how it goes, thanks for the tip

Answer 8

Your welcome.

Answer 9

Well, that was easy. Thanks again