There hasn’t been a lot to blog about recently, so this is another calculus post. Sorry to the math dislikers…
I’ve reached the second last chapter of my Calculus 1 self studying journey, which inevitably includes a fundamental part of calculus, the indefinite integral. I’m not particularly deep into the topic, but it’s surprising how much of this you can relate to basic physics, even with a simple understanding. The lesson from Professor Leonard that I watched today includes an example where you must solve for the max height and time of flight for a projectile, using only initial speed and vertical displacement:
A catapult accidentally shoots straight up with an initial velocity of 128 ft/s, from a height of 16 ft. Find:
- The position function for the height, s(t)
- The maximum height of the projectile
- The time it takes the projectile to hit the ground
For a physics class, this would be an absurdly easy question, but the aim in this context was to keep taking the antiderivative of acceleration (-g) until you’d end up with a function that gives you position (s):
I originally considered this problem as being pretty different than anything in a physics class, especially since you don’t really deal with any functions in physics, every variable you’ll deal with will be compacted into a simple formula. This was until I realized something: To solve this problem, you are literally doing the exact same thing you’d do in a physics problem. You are simply deriving the formula first, rather than just finding it off of your formula sheet.
This was a rather unexpected epiphany for me. For a long time I’ve just memorized the kinematics equations without even knowing how simple it was to see where they came from. I know that there’s a more straightforward, algebraic way to prove this using the area under a graph, but I really do think the calculus is what ties everything together. In the past I’ve viewed physics and mathematics being two separate entities for whatever reason, and this is what really bridged the gap for me.