Why is the unit of specific impulse in seconds, and how does it wind up being proportional to exhaust velocity, which seems like it should be kg*m/s ?
Summarizing the helpful links people have posted: The starting point is impulse, which is how much 'push' you can get out of your fuel, which is force*time. For the same fuel you can do less force for longer, or more force for shorter, but the product is the same \
I for one have a tendency to call this 'power' which is incorrect, it's 'impulse'. Since force = mass*acceleration and impulse = force*time, impulse = kg*(m/s*s)*s = kg*(m/s) \
This is then divided by the weight it's working against: specific impulse is different on different planets! weight is mass*gravity, which for earth is kg*9.8*m/(s*s). That constant is different on the moon! \
Dividing impulse by weight you get (kg*m/s)/(kg(m/(s*s))) which cancels out to seconds. So that's why the units are in seconds.
Other fun units things to wrap your brain around: time = resistance * capacitance and vehicle energy consumption is measured in square millimeters
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