Linear Motion

Newton's Laws and Conservation Laws¶

Elastic Collision in 1D¶

The coefficient of restitution is \(\(e = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}\)\)

When \(e = 1\), the collision is perfectly elastic collision. When \(0 \le e \lt 1\), the collision is inelastic collision. When \(e = 0\), the collision is perfectly inelastic collision.

In the perfectly elastic collision,

\[ \begin{gathered} v_{1f} = \frac{m_1-m_2}{m_1+m_2} v_{1i} + \frac{2m_2}{m_1+m_2} v_{2i} = 2v_{CM} - v_{1i} \\\\ v_{2f} = \frac{2m_1}{m_1+m_2} v_{1i} + \frac{m_2 - m_1}{m_1+m_2} v_{2i} = 2v_{CM} - v_{2i} \end{gathered} \]

When \(m_1 >> m_2\), we can get \(v_{1f} = v_{1i}\), \(v_{2f} = 2v_{1i} - v_{2i}\). When \(m_1 = m_2\), we can get \(v_{1f} = v_{2i}\), \(v_{2f} = v_{1i}\).

Elastic Collision in 2D¶

\[ \begin{cases} m_1v_{1f}\cos\theta + m_2v_{2f}\cos\phi = m_1v_{1i} \\ m_1v_{1f}\sin\theta - m_2v_{2f}\sin\phi = 0 \\ \displaystyle \frac{1}{2}m_1v_{1i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 \end{cases} \]

When \(m_1 = m_2\) and \(v_{2i} = 0\), we can get \(\boldsymbol{v}_{1f} \cdot \boldsymbol{v}_{2f} = 0\)

Center of Mass¶

\[x_\text{CM} = \lim_{\Delta m_i \rightarrow 0} \frac{\sum x_i \Delta m_i}{M} = \frac{1}{M} \int x dm\]

\[\boldsymbol{r}_\text{CM} = \frac{\sum m_i \boldsymbol{r}_i}{M}\]

\[\boldsymbol{v}_\text{CM} = \frac{\sum m_i \boldsymbol{v}_i}{M}\]

\[\boldsymbol{a}_\text{CM} = \frac{\sum m_i \boldsymbol{a}_i}{M}\]

\[\boldsymbol{p}_\text{CM} = \sum m_i\boldsymbol{v}_i = M\boldsymbol{v}_\text{CM}\]

The kinetic energy of a system is equal to the kinetic energy of its center of mass plus its kinetic energy in the CM frame.

\[ \begin{aligned} E_k &= \sum \frac{1}{2} m_i (v_i' + v_\text{CM})(v_i' + v_\text{CM}) \\ &= \sum \frac{1}{2} m_i v_\text{CM}^2 + \sum m_i v_i' v_\text{CM} + \sum \frac{1}{2} m_i v_i'^2 \\ &= \frac{1}{2} v_\text{CM}^2 \sum m_i + \sum \frac{1}{2} m_i v_i'^2 + v_\text{CM} \sum m_i v_i' \\ &= \frac{1}{2} v_\text{CM}^2 M + \sum \frac{1}{2} m_i v_i'^2 + v_\text{CM}\frac{d}{dt} \sum m_i r_i' \\ &= E_{Kcm} + E_k' \end{aligned} \]

Perfectly Elastic Collision in CM Frame¶

\[ \begin{gathered} v_{1i}' = v_{1i} - v_\text{CM} = \frac{m_2}{m_1+m_2} (v_{1i} - v_{2i}) \\\\ v_{2i}' = v_{2i} - v_\text{CM} = \frac{m_1}{m_1+m_2} (v_{2i} - v_{1i}) \\\\ v_{1f}' = -v_{1i}' \\\\ v_{2f}' = -v_{2i}' \end{gathered} \]

Perfectly inelastic scattering in CM Frame¶

\[ \begin{gathered} v_{1i}' = v_{1i} - v_\text{CM} = \frac{m_2}{m_1+m_2} (v_{1i} - v_{2i}) \\\\ v_{2i}' = v_{2i} - v_\text{CM} = \frac{m_1}{m_1+m_2} (v_{2i} - v_{1i}) \\\\ v_{1f}' = v_{2f}' = 0 \end{gathered} \]

Gravitation¶

The Law of Gravitation¶

The law of gravitation is \(\(F = G \frac{Mm}{R^2}\)\)

So the period at an circular orbit is

\[ \begin{gathered} \frac{GM}{R^2} = \omega^2 R = \frac{4\pi^2}{T^2}R \\\\ \Rightarrow \frac{T^2}{R^3} = \frac{4\pi^2}{GM} \\\\ \Rightarrow T = 2\pi\sqrt{\frac{R^3}{GM}} \end{gathered} \]

And the corresponding speed is

\[ \begin{gathered} \frac{GM}{r^2} = \frac{v^2}{r} \\\\ \Rightarrow v = \sqrt{\frac{GM}{r}} \end{gathered} \]

Then we can get the energy of the satellite \(\(E = \frac{1}{2}mv^2 - \frac{GMm}{r} = -\frac{GMm}{2r}\)\)

Rocket Propulsion¶

The first cosmic speed is \(v_1 = \sqrt{\frac{GM}{R}}\).

The second cosmic speed is \(v_1 = \sqrt{\frac{2GM}{R}}\).

Let \(m\) be the mass of the ejected medium in an instant, \(v_e\) the speed of the medium relative to the rocket, then we can get

\[ \begin{gathered} M dv = v_e dm = -v_e dM \\\\ \Rightarrow dv = -v_e \frac{dM}{M} \\\\ \Rightarrow v_f - v_i = v_e \ln \frac{M_i}{M_f} \end{gathered} \]

Plus, taking the gravity into consideration, we can get

\[ \begin{gathered} Ma = -v_e \frac{dM}{M} - gt \\\\ \Rightarrow adt = v_e \frac{dM}{M} - gdt \\\\ \Rightarrow v = v_e \ln(\frac{M_0}{M}) - gt \end{gathered} \]