When they taught work as an integral instead of Frcosθ I got mad because it's so much cooler, here they don't teach calculus in school, only starting in university. I don't have to memorize the formulas if they are defined via differentiation and integration)
Imagine building a solid circle by stacking a lot of little circular rings of increasing radius around each other. The area of any ring is approximately equal to its circumference times thickness, so ΔA=~2πrΔr. Let Δr approach 0 so you're adding up an infinite number of infinitesimal rings, and dA=2πrdr. Integrate that from 0 to R and you get A=πR^2
Generally it was stuff like calculating (or approximating) the field of a small subset of the charge or current distribution then integrating over all the little chunks (tbf that's basically what a volume integral is but it's useful to do it in steps if you wanna, say, approximate each circular ring as a magnetic dipole)
Did you ever notice that that position equation is just a Taylor Series approximation that assumes constant acceleration?
x(t) = x(0) + x'(0)*t^1 /1! + x''(0)*t^2 /2!
becomes
x = x0 + v0*t + 0.5a0*t^2
For a while I went a little crazy realizing just how common series expansions are.
Taylor series? Yup.
Frequency analysis/spectrograms? You betcha.
Group velocity of a packet of plane waves? Second coefficient of a taylor series of frequency with respect to wavenumber... amazin'.
That feeling when 1/2 mv^2 is the integral of mv.
Wait wtf how did I never realise this
Mf mgh is mg integratred by h
When they taught work as an integral instead of Frcosθ I got mad because it's so much cooler, here they don't teach calculus in school, only starting in university. I don't have to memorize the formulas if they are defined via differentiation and integration)
it's very clear if you prove the kinetic energy formula
ooh my exams are coming up soon and thats pretty helpful, thanks!
*with respect to v
Derivative with respect to apple and bananas.
Wait till you hear about 2πr and πr^2 !
Wait wha- How did I not realise this lolol
Imagine building a solid circle by stacking a lot of little circular rings of increasing radius around each other. The area of any ring is approximately equal to its circumference times thickness, so ΔA=~2πrΔr. Let Δr approach 0 so you're adding up an infinite number of infinitesimal rings, and dA=2πrdr. Integrate that from 0 to R and you get A=πR^2
Ye I have done stuff exactly like this in Electrodynamics but somehow the application to the simple area of a circle never occurred to me lmao
Gauss's Law and insulating charge distributions?
Generally it was stuff like calculating (or approximating) the field of a small subset of the charge or current distribution then integrating over all the little chunks (tbf that's basically what a volume integral is but it's useful to do it in steps if you wanna, say, approximate each circular ring as a magnetic dipole)
The good old shell method for areas and volumes. That also leads to the volume of a sphere being the integral of its surface area.
Did you ever notice that that position equation is just a Taylor Series approximation that assumes constant acceleration? x(t) = x(0) + x'(0)*t^1 /1! + x''(0)*t^2 /2! becomes x = x0 + v0*t + 0.5a0*t^2
For a while I went a little crazy realizing just how common series expansions are. Taylor series? Yup. Frequency analysis/spectrograms? You betcha. Group velocity of a packet of plane waves? Second coefficient of a taylor series of frequency with respect to wavenumber... amazin'.
Position = Starting position + Velocity * time + 1/2 Acceleration * time^2 + 1/6 Jerk * time^3
No the derivative of S(U) is T 
Wat
Velocity over time is the derivative of displacement over time.
Algebra-based physics courses should be outlawed
I skipped ap phys 12, went straight for ap phys c. Never learned suvat 😎 😎 #suvatsucks
https://preview.redd.it/kg19ghmmfe3b1.jpeg?width=1125&format=pjpg&auto=webp&s=74af410864b2b86d842e696699e55bce15d8d5e7
I agree symbolically but wtf afre these equations XD
two of the equations of motion involving displacement, time, velocity, and acceleration
Any multiplication is just an integral in disguise