- #1

Mr Davis 97

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There are two different ways I have seen. First we start with ##\displaystyle \frac {dv}{dt} = a##. Next, we multiply both sides by velocity, ##\displaystyle v\frac {dv}{dt} = a\frac {dx}{dt} ##. From here are where the two ways diverge. One author "cancels" out both ##dt## terms and proceeds to integrate with respect to the differential on each side (dv and dx). However, how is this justified? Doesn't one need to always introduce a variables of integration to the equation when they integrate both sides? I don't see how it's justified to just use the differentials that are already there. Another author does it a different way. He takes ##\displaystyle v\frac {dv}{dt} = a\frac {dx}{dt} ## and integrates both sides with respect to time. However, I think it gets fishy when he cancels out the ##dt## in the derivatives with the ##dt## that he introduces as a variable of integration, rendering the variables of integration for each side as ##v## and ##x##. Also, the bounds of integration are ##t_{1}## and ##t_{2}## for both integrals, so how do the bounds of integration somehow become in terms of ##x## and ##v## respectively? If someone could answer my questions for both cases, I would be happy.