# Physics

## Motion

### Acceleration

$$
a = \frac{v_f - v_i}{\Delta t}
$$

- $v_f$: final velocity
- $v_i$: initial velocity

Unit: $m/s^2 = (m/s) /s$

### Motion Formulas

$$
\begin{cases}
v = v_0 + at \\
d = v_0 t + \frac{1}{2} a t^2 \\
v^2 = v_0^2 + 2ad
\end{cases}
$$

- $v$: velocity
- $v_0$: initial velocity

## Newton's Laws

Newton's second law:

$$
\vec{F} = m \vec{a}
$$

## Spring

Hooke's law:

$$
\vec{F}_s = -k \vec{x}
$$

Energy:

$$
E_s = \frac{1}{2} k x^2
$$

## Work / Energy

Work:

$$
W = \vec{F} \cdot \vec{S}
$$

Kinetic energy:

$$
E_K = \frac{1}{2} m v^2
$$

Potential energy:

$$
U_g = mgh
\qquad
U_s = \frac{1}{2} k x^2
$$

**Conservation of energy** holds when there are only **conservative forces**.

$$
\Delta U + \Delta K = 0
$$

$$
\Delta E = \Delta U + \Delta K = W_{\text{non-conservative}}
$$

## Momentum

$$
\vec{p} = m \vec{v}
$$

$$
\vec{F} = m \vec{a} = m \frac{\Delta \vec{v}}{\Delta t}
= \frac{\Delta \vec{p}}{\Delta t}
$$

$$
F = \frac{\Delta p}{\Delta t}
$$

Initial: $p_{\text{total}} = m v_1 + m \cdot 0 = m v_1$
After collision:
$p_{\text{total}} = m v_1 = m v_2$

$$
\implies v_1 = v_2
$$

---

Initial:
$p_{\text{total}} = m_1 v_1 + m_2 v_2$
After collision:
$p_{\text{total}} = m_1 v_1' + m_2 v_2'$

$$
m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'
$$

---

$$
2 \cdot 1 + 0 = -2v_1' + 4 v_2'
$$

$$
-v_1' + 2 v_2' = 1
$$

Initial energy:
$\frac{1}{2} m_1 v_1^2 = 1$
After collision:
$\frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2$

$$
\frac{1}{2} m_1 v_1^2 = \frac{1}{2} m_1 v_1'^2 + \frac{1}{2} m_2 v_2'^2
$$

$$
v_1'^2 + 2 v_2'^2 = 1
$$

$$
\begin{cases}
-v_1' + 2 v_2' = 1 \\
v_1'^2 + 2 v_2'^2 = 1
\end{cases}
$$

$$
v_1' = 2v_2' - 1
$$

$$
(2v_2' - 1)^2 + 2 v_2'^2 = 1
$$

$$
6v_2'^2 - 4 v_2' = 0
$$

$$
v_2'(3v_2' - 2) = 0
$$

$$
v_2' = 0, \frac{2}{3}
\qquad
v_1' = -1, \frac{1}{3}
$$

Impulse:

$$
I = F \cdot \Delta t = \Delta p
$$

## Circular Motion

- $T$ : period (s)
- $f$ : frequency (circle/s)
- $\omega$ : angular velocity (rad/s)
- $\alpha$ : angular acceleration
- $a_c$ : centripetal acceleration (m/s$^2$)
- $F_c$ : centripetal force (N)

$$
\omega = \frac{\Delta \theta}{\Delta t}
= \frac{v}{r}
\qquad
v = r \omega
$$

$$
T = \frac{2 \pi}{\omega}
$$

$$
a_c = \frac{v^2}{r}
= r \omega^2
= \frac{4 \pi^2 r}{T^2}
$$

$$
F_c = \frac{mv^2}{r}
= mr \omega^2
= \frac{4 \pi^2 mr}{T^2}
$$

- $L$ : angular momentum
- $I$ : moment of inertia, angular mass
- $\tau$ : torque （力矩）

$$
L = rmv = r^2 m \omega = I \omega
$$

$$
I = \frac{L}{\omega}
= \frac{rmv}{\omega}
= \frac{rmv}{\frac{v}{r}}
= mr^2
$$

$$
\vec{\tau} = \vec{F} \times \vec{r}
$$

Newton's second law for rotation:

$$
\tau = I \alpha = I \frac{\Delta \omega}{\Delta t} = \frac{\Delta L}{\Delta t}
$$

## Simple Harmonic Motion


$$
y = A \cos(\omega t) = A \cos(2 \pi ft)
$$

$$
F = kA = \frac{4 \pi^2 mr}{T^2}
$$

$$
T^2 = \frac{4 \pi^2 m r}{kA} = \frac{4 \pi^2 m}{k}
$$

$$
T = 2\pi \sqrt{\frac{m}{k}}
$$

$$
f = \frac{1}{T} = \frac{1}{2\pi} \sqrt{\frac{k}{m}}
$$

Pendulum:

$$
T = 2\pi \sqrt{\frac{L}{g}}
$$

## Electrostatics

$$
F = \frac{kQq}{r^2}
$$

$$
E = \frac{F}{q} = \frac{kQ}{r^2}
$$

## Electric Current

$$
I = \frac{q}{t}
$$

Ohm's law:

$$
V = IR
$$

$$
R = \frac{\rho L}{A}
$$

Parallel resistance:

$$
\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots
$$

## Misc.

$$
d = v_0 t + \frac{1}{2} a t^2
$$

$$
56 = 8 t + \frac{1}{2} \cdot 3 \cdot t^2
$$

$$
112 = 16 t + 3 t^2
$$

$$
3 t^2 + 16 t - 112 = 0
$$

---

$$
6 = -v_1 + 2 v_2
$$

$$
18 = \frac{1}{2} v_1^2 + v_2^2
$$

$$
v_1 = 2 v_2 -6
$$

$$
18 = \frac{1}{2} (2 v_2 -6)^2 + v_2^2
$$

$$
18 = (2 v_2^2 - 12 v_2 + 18) + v_2^2
$$

$$
\implies v_2 = 4, v_1 = 2
$$

$$
12 = -v_1 + 3 v_2
$$

$$
72 = \frac{1}{2} v_1^2 + \frac{3}{2} v_2^2
$$

$$
100 = 50 v_1, v_1 = 2
$$

$$
10 = \frac{1}{2} \cdot 2.5 \cdot t^2
$$

$$
t^2 = 8, \; t = 2\sqrt{2}
$$

$$
a = 2.5
$$

$$
v = at = 2.5 \cdot 2\sqrt{2}
$$

$$
2 = \frac{1}{2} 0.01 v^2
$$

$$
v^2 = 400
$$

$$
\frac{9 \cdot 10^9 \cdot 5 \cdot 10^{-6} \cdot 6 \cdot 10^{-6}}{400}
= \frac{270}{400} \cdot 10^{-3}
= \frac{27}{4} \cdot 10^{-4}
$$