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Momentum

Linear momentum​

SI unit: kg⋅\cdotm/s

p⃗=mv⃗F⃗net=dp⃗dt\begin{align*} \vec p &= m \vec v \\ \vec F_{net} &= \frac{d\vec p}{dt} \end{align*}

Impulse​

SI unit: N⋅\cdots

J⃗=F⃗netΔt=Δp⃗=∫t1t2F⃗netdt\begin{align*} \vec J &= \vec F_{net} \Delta t = \Delta\vec p \\ &= \int_{t_1}^{t_2}\vec F_{net} dt \end{align*}

Angular Momentum​

SI unit: kg⋅\cdotm2^2/s

L⃗=r⃗×p⃗τ⃗=dL⃗dt=r⃗×F⃗L=mvrsin⁡θL=JdL=Iω\begin{align*} \vec L &= \vec r \times\vec p \\ \vec \tau &= \frac{d\vec L}{dt} = \vec r \times\vec F \\ L &= mvr\sin\theta \\ L &= Jd \\ L &= I\omega \end{align*}
  • L=IωL = I\omega is true only if L⃗\vec L is along the rotation axis
  • Directions of ω⃗\vec\omega and L⃗\vec L are the same, defined by the right-hand thumb rule

Collisions​

  • Total momentum is conserved
  • Elastic collision: KE and total energy conserved
  • Inelastic collision: KE decreases during collision
  • Completely inelastic collision: Objects stick together and KE decreases greatly during collision

Elastic collision​

vA0−vB0=−(vA1−vB1)\begin{align*} v_{A0} - v_{B0} = -(v_{A1} - v_{B1}) \end{align*}
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Special case: if object B is initially at rest

vA1=(mA−mBmA+mB)vA0vB1=(2mAmA+mB)vA0\begin{align*} v_{A1} &= \left(\frac{m_A - m_B}{m_A + m_B}\right) v_{A0} \\ v_{B1} &= \left(\frac{2m_A}{m_A + m_B}\right) v_{A0} \end{align*}