期中：8.29 14:00-16：00chapter 1-13 & 30-31
v = v 0 + ∫ 0 t a d t v = v_0+\int_0^t a dt v = v 0 + ∫ 0 t a d t x = x 0 + ∫ 0 t v d t x = x_0+\int_0^t vdt x = x 0 + ∫ 0 t v d t
Vector
Scalar
Magnitude & Direction
Only magnitude
motions in conponents: 3 independent motions in each demension
projectile motion
uniform circular motion
angular velocityω = d θ d t = 1 r d s d t = v r \omega = \frac{d\theta}{dt}=\frac{1}{r}\frac{ds}{dt}=\frac{v}{r} ω = d t d θ = r 1 d t d s = r v
angular acceleration α = d ω d t \alpha = \frac{d\omega}{dt} α = d t d ω
radial acceleration: direction → \rightarrow → v ⃗ = v τ ⃗ \vec{v}=v\vec{\tau} v = v τ a ⃗ = d v ⃗ d t = d v d t τ + v d τ ⃗ d t = v lim Δ t → 0 Δ τ ⃗ Δ t \vec{a} =\frac{d\vec{v}}{dt}=\frac{dv}{dt}\tau+v\frac{d\vec{\tau}}{dt}=v\lim_{\Delta t\rightarrow0}\frac{\Delta\vec{\tau}}{\Delta{t}} a = d t d v = d t d v τ + v d t d τ = v Δ t → 0 lim Δ t Δ τ Δ t → 0 \Delta t\rightarrow0 Δ t → 0 Δ τ ⃗ → Δ θ n ⃗ \Delta \vec{\tau} \rightarrow\Delta\theta \vec{n} Δ τ → Δ θ n d v ⃗ d t = v lim Δ t → 0 Δ τ ⃗ Δ t = v d θ d t n ⃗ = v 2 r n ⃗ = a R ⃗ \frac{d\vec{v}}{dt}=v\lim_{\Delta t\rightarrow0}\frac{\Delta\vec{\tau}}{\Delta{t}}=v\frac{d\theta}{dt}\vec{n} = \frac{v^2}{r}\vec{n}=\vec{a_R} d t d v = v Δ t → 0 lim Δ t Δ τ = v d t d θ n = r v 2 n = a R
a R ⃗ \vec{a_R} a R
Nonuniform circular motion:a ⃗ = d v ⃗ d t = d v d t τ ⃗ + v 2 r n ⃗ = a t a n ⃗ + a R ⃗ \vec{a} =\frac{d\vec{v}}{dt}=\frac{dv}{dt} \vec{\tau}+\frac{v^2}{r}\vec{n}=\vec{a_{tan}}+\vec{a_R} a = d t d v = d t d v τ + r v 2 n = a t a n + a R
a t a n ⃗ \vec{a_{tan}} a t a n
a R ⃗ \vec{a_R} a R
Properties of acceleration
a t a n ⃗ ⊥ a R ⃗ \vec{a_{tan}}\bot \vec{a_R} a t a n ⊥ a R a = a t a n 2 ⃗ + a R 2 ⃗ a=\sqrt{\vec{a^2_{tan}}+ \vec{a^2_R}} a = a t a n 2 + a R 2
t a n φ = a R ⃗ a t a n ⃗ tan\varphi = \frac{\vec{a_R}}{\vec{a_{tan}}} t a n φ = a t a n a R
a t a n = α r {a_{tan}} = \alpha r a t a n = α r