Video 2: Now with Calculus

Deriving Kinematics Equations

Change in Velocity

a⃗=dvdt\vec a = \frac{dv}{dt}
a⃗∗dt=dv\vec a * dt = dv
∫0tadt=∫vivfdv\int_0^t a dt = \int_{v_i}^{v_f} dv
at∣0t=v∣vivfat |_0^t = v|_{v_i}^{v_f}
at−a(0)=vf−viat - a(0) = v_f - v_i
vf=vi+atv_f = v_i + at

Change in Position

v=vi+atv = v_i + at
dxdt=vi+at\frac{dx}{dt} = v_i + at
dx=(vi+at)dtdx = (v_i + at)dt
∫xixfdx=∫0t(vi+at)dt\int_{x_i}^{x_f} dx = \int_0^t (v_i + at)dt
x∣xixf=(vit+12at2)∣0tx|_{x_i}^{x_f} = (v_it + \frac{1}{2} a t^2)|_{0}^{t}
xf−xi=(vit+12at2)−0x_f - x_i = (v_i t + \frac{1}{2}at^2) - 0
Δx=vit+12at2\Delta x = v_i t + \frac{1}{2}at^2

Bonus Kinematics Equation

a⃗=dvdt\vec a = \frac{dv}{dt}
a⃗=dvdxdxdt\vec a = \frac{dv}{dx} \frac{dx}{dt}
a⃗=v⃗dvdx\vec a = \vec v \frac{dv}{dx}
∫xixfa⃗dx=∫v0vfvdv\int_{x_i}^{x_f} \vec a dx = \int_{v_0}^{v_f} v dv
ax∣xixf=12v2∣v0vfax |_{x_i}^{x_f} = \frac{1}{2} v^2 |_{v_0}^{v_f}
2aΔx=vf2−vi22 a \Delta x = v_f^2 - v_i^2
vf2=vi2+2aΔxv_f^2 = v_i^2 + 2a\Delta{x}