We recently started going over momentum in my phyics class and we were given this bit of information: ##impulse=FΔt=mΔv=momentum## I noticed that I were to take the integral of both sides with respect to velocity, it would yield ##FvΔt=Fx=(1/2)mv^2## Which is the Work-Kinectic energy relationship. At first, I thought that integrating momentum and getting Kinetic Energy was just a coincidence, but then after thinking about how momentum and kinetic energy could be related and seeing that integrating both sides still yields a familiar formula I realize that it has to mean something. However, I can't figure it out, and my professor was unable to give me an answer. So, is this relationship similiar to the relationship between Acceleration, Velocity, and Position, where, rather than integrating with respect to time we are integrating with respect to position and time? How am I to interpret this? i.e. Acceleration can be described as the change in velocity over some time and the velocity, the change in position over some time. Differntial equations comes to mind when I think about integrating with respect to two variables at once, but alas, the answer eludes me! I've been chewing on this for days, so any insights are greatly appreciated!