Newton's Laws with Calculus

Newton's Second Law

Vector form: $\vec{F}_{net} = m\vec{a} = m\frac{d\vec{v}}{dt} = m\frac{d^2\vec{r}}{dt^2}$

Component form: $F_x = m\frac{dv_x}{dt}, \quad F_y = m\frac{dv_y}{dt}, \quad F_z = m\frac{dv_z}{dt}$

Impulse-Momentum Theorem

From Newton's second law: $\vec{F} = \frac{d\vec{p}}{dt}$

where momentum $\vec{p} = m\vec{v}$.

Integrating both sides: $\int_{t_1}^{t_2} \vec{F} \, dt = \int_{t_1}^{t_2} \frac{d\vec{p}}{dt} \, dt = \vec{p}_2 - \vec{p}_1 = \Delta \vec{p}$

Impulse: $\vec{J} = \int_{t_1}^{t_2} \vec{F} \, dt = \Delta \vec{p}$

Variable Force Example

If $F(t) = F_0\cos(\omega t)$ for time $0$ to $T = \pi/(2\omega)$:

$J = \int_0^T F_0\cos(\omega t) \, dt = \frac{F_0}{\omega}\sin(\omega t)\Big|_0^T = \frac{F_0}{\omega}$

Variable Mass Systems

For systems where mass changes with time (rockets):

$\vec{F}_{ext} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt} = m\frac{d\vec{v}}{dt} + \vec{v}\frac{dm}{dt}$

Rocket equation: $m\frac{dv}{dt} = -v_{rel}\frac{dm}{dt} + F_{ext}$

where $v_{rel}$ is exhaust velocity relative to rocket.

For no external forces and integrating: $\Delta v = v_{rel}\ln\frac{m_0}{m_f}$

(Tsiolkovsky rocket equation)

Drag Forces

Linear drag: $F_d = -bv$

$m\frac{dv}{dt} = -bv$

Solution: $v(t) = v_0e^{-bt/m}$

Quadratic drag: $F_d = -cv^2$

$m\frac{dv}{dt} = -cv^2$

Separating variables: $\int_{v_0}^v \frac{dv'}{v'^2} = -\frac{c}{m}\int_0^t dt'$

$v(t) = \frac{v_0}{1 + \frac{cv_0t}{m}}$

Falling with Air Resistance

Equation of motion: $m\frac{dv}{dt} = mg - bv$

Terminal velocity: $v_t = \frac{mg}{b}$

Rewrite as: $\frac{dv}{dt} = g - \frac{b}{m}v = \frac{b}{m}(v_t - v)$

Solution: $v(t) = v_t(1 - e^{-bt/m})$

Forces in Polar Coordinates

Newton's second law in polar coordinates:

Radial direction: $F_r = m(\ddot{r} - r\dot{\theta}^2)$

Tangential direction: $F_{\theta} = m(r\ddot{\theta} + 2\dot{r}\dot{\theta})$

Example: Central Force

For motion under central force (depends only on $r$):

$F_{\theta} = 0$, so: $r\ddot{\theta} + 2\dot{r}\dot{\theta} = 0$

$\frac{d}{dt}(r^2\dot{\theta}) = 0$

This means $r^2\dot{\theta} = L/m$ is constant (angular momentum conservation).