Today's lecture was a review for the first midterm, coming up! Most of the lecture was spent on going over problems, which I won't reproduce here: refer to the recording and slides on Canvas. However, I will reproduce my condensed list of physics formulas here.
(Note: this list focuses on physics, not math. I give no guarantee that every single formula you'll need is on the list! Making your own formula sheet with whatever information you find useful would be a good way to study.)
Newton's three laws:
These are true in inertial (non-accelerating) reference frames.
Important curvilinear coordinates:
Cartesian: \( (x,y,z) \)
Cylindrical: \( (\rho, \phi, z) \)
Newton's second law in cylindrical components:
\[ \begin{aligned} F_\rho = m(\ddot{\rho} - \rho \dot{\phi}^2) \\ F_\phi = m(\rho \ddot{\phi} + 2\dot{\rho} \dot{\phi}) \end{aligned} \]
Spherical: \( (r, \theta, \phi) \)
Drag force is velocity-dependent. Most general form: \( \vec{f}(\vec{v}) = -f(v) \hat{v} \).
For motion small compared to the speed of sound, \( f(v) = bv + cv^2 \).
Microscopic origin:
For an object with linear size \( D \), and a medium with viscosity \( \eta \),
\( f_{\rm lin}(v) = 3\pi \eta D v \), so \( b = 3\pi \eta D \).
Shortcut formula (air at STP): \( b = \beta D \), \( \beta = 1.6 \times 10^{-4}\ {\rm N} \cdot {\rm s} / {\rm m}^2 \).
For an object with cross-sectional area \( A \) and drag coefficient \( C_D \), and a medium with density \( \rho \),
\( f_{\rm quad}(v) = \frac{1}{2} C_D \rho A v^2 \), so \( c = \frac{1}{2} C_D \rho A \).
Shortcut formula (sphere in air at STP): \( c = \gamma D^2 \), \( \gamma = 0.25\ {\rm N} \cdot {\rm s}^2 / {\rm m}^4 \).
Usually we can ignore either linear or quadratic drag, if one is much larger. Determine relative size through Reynolds number,
\[ \begin{aligned} R = \frac{f_{\rm quad}}{f_{\rm lin}} = \frac{D\rho v}{\eta}. \end{aligned} \]
\( R \ll 1 \): keep linear force. \( R \gg 1 \): keep quadratic force.
Horizontal motion, no other forces:
\[ \begin{aligned} v_x(t) = v_0 e^{-t/\tau}, \\ \end{aligned} \]
where natural time \( \tau = m/b \).
Vertical motion w/gravity (note that \( +y \) is up):
\[ \begin{aligned} v_y(t) = v_0 e^{-t/\tau} - v_{\rm ter} (1 - e^{-t/\tau}), \end{aligned} \]
where terminal velocity \( v_{\rm ter} = mg/b = g\tau \).
Horizontal motion, no other forces:
\[ \begin{aligned} v_x(t) = \frac{v_0}{1+t/\tau_c}, \end{aligned} \]
where quadratic natural time \( \tau_c = m/(cv_0) \).
Vertical motion w/gravity (note that \( +y \) is up, assume \( v_y < 0 \)):
\[ \begin{aligned} v_y(t) = -v_{\rm ter,c} \tanh \left( \frac{t}{\tau} \right) \end{aligned} \]
where quadratic terminal velocity \( v_{\rm ter,c} = \sqrt{mg/c} \) and \( \tau = v_{\rm ter,c}/g \).
Conservation of linear momentum: for a collection of \( N \) masses \( {m_\alpha} \),
\[ \begin{aligned} \vec{F}_{\rm net, ext} = \frac{d\vec{P}}{dt} \end{aligned} \]
where
\[ \begin{aligned} \vec{P} = \sum_\alpha \vec{p}_\alpha = \sum_{\alpha=1}^N m_\alpha \dot{\vec{r}}_\alpha. \end{aligned} \]
No net external force implies \( \vec{P} \) is constant.
Net external force acts on the center of mass (CM) like a point particle,
\[ \begin{aligned} \vec{F}_{\rm net, ext} = M \ddot{\vec{R}}. \end{aligned} \]
Total mass:
\[ \begin{aligned} M = \sum_\alpha m_\alpha \end{aligned} \]
or continuous version with density function \( \rho(\vec{r}) \):
\[ \begin{aligned} M = \int dm = \int \rho(\vec{r}) dV. \end{aligned} \]
CM position:
\[ \begin{aligned} \vec{R} = \sum_\alpha m_\alpha \vec{r}_\alpha \end{aligned} \]
or continuous version,
\[ \begin{aligned} \vec{R} = \int \vec{r} dm = \int \rho(\vec{r}) \vec{r}\ dV. \end{aligned} \]
Reflection symmetry: contributions by equal masses on opposite sides of a coordinate axis, e.g. \( (x,y,z) \) and \( (-x,y,z) \), cancel out for CM position. Complete reflection symmetry in a direction --> that CM coordinate is zero.
Rocket equation of motion:
\[ \begin{aligned} m \frac{dv}{dt} = F_{\rm thrust},\ \ F_{\rm thrust} = -|v_{\rm exh}| \frac{dm}{dt} \end{aligned} \]
with fuel exhaust speed \( |v_{\rm exh}| \) (always positive) and mass ejection rate \( dm/dt \) (always negative!) If other forces are present, add alongside \( \vec{F}_{\rm thrust} \) as normal.
Speed-mass relationship:
\[ \begin{aligned} v(m) = v_0 + |v_{\rm exh}| \ln(m_0/m) \end{aligned} \]
If initial mass \( m_0 = m_E + m_F \), where \( m_F \) is fuel mass and \( m_E \) is empty mass, then max delta-v from burning all fuel is
\[ \begin{aligned} \Delta v_{\rm max} = |v_{\rm exh}| \ln \left( 1 + \frac{m_F}{m_E} \right). \end{aligned} \]