Theoretical Mechanics IPSP

Jürgen Vollmer, Universität Leipzig

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book:chap5:5.2_collisions_of_particles [2022/01/04 03:27] – [5.2.3 The collision] abrilbook:chap5:5.2_collisions_of_particles [2022/01/04 05:12] (current) abril
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-FIXME draft with missing figures and references :!: 
- 
 ===== 5.2  Collisions of hard-ball particles  ===== ===== 5.2  Collisions of hard-ball particles  =====
 <WRAP id=section_particleScattering /> <WRAP id=section_particleScattering />
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 ==== 5.2.4 Self Test ==== ==== 5.2.4 Self Test ====
  
-<WRAP right id=fig_HardBallScatteringAngle> +<WRAP 140pt right #fig_HardBallScatteringAngle> 
-{{./Sage/EOM_HardBallScattering__TrajectoryShape.png}} +{{EOM_HardBallScattering.png}}
-{{./Sage/EOM_HardBallScattering__ScatteringAngle.png}}+
  
-Collision of two hard-ball particles with radii $R_1$ and $R_2$:+Figure 5.7: Collision of two hard-ball particles with radii $R_1$ and $R_2$:
 (top) Trajectory shape. The labels denote the ratios $(\mathbf p \cdot \hat{\boldsymbol \alpha}) / (\mathbf p \cdot \hat{\boldsymbol \beta})$. (top) Trajectory shape. The labels denote the ratios $(\mathbf p \cdot \hat{\boldsymbol \alpha}) / (\mathbf p \cdot \hat{\boldsymbol \beta})$.
-(bottom) Scattering angle $\theta$. }+(bottom) Scattering angle $\theta$.
 </WRAP> </WRAP>
- 
-----  
  
 <wrap #quest_HardBallScatteringAngle > Problem 5.2: </wrap>** Scattering angle for hard-ball particles ** <wrap #quest_HardBallScatteringAngle > Problem 5.2: </wrap>** Scattering angle for hard-ball particles **
 \\ \\
-In\cref{fig:HardBallScatteringAngle} we show shows the trajectory shape and the scattering angle for hard-ball scattering.+In [[#fig_HardBallScatteringAngle |Figure 5.7]] we show shows the trajectory shape and the scattering angle for hard-ball scattering.
  
-  -  What is the dimensionless length scale adopted to plot the trajectory shapes? +**a)** What is the dimensionless length scale adopted to plot the trajectory shapes?\\ 
-  -  What is the impact of the angular momentum on the trajectory shape? + 
-\\ +**b)** What is the impact of the angular momentum on the trajectory shape? What is the impact of the energy?\\ 
-What is the impact of the energy? + 
-  -  Verify that  +**c)** Verify that  
-\begin{align}  \label{eq:HardBallScatteringAngle+<wrap #eq_HardBallScatteringAngle></wrap> 
-      \sin^2\theta = \frac{ L^2 }{2\mu \, E \, (R_1+R_2)^2}+\begin{align}  
 +      \sin^2\theta = \frac{ L^2 }{2\mu \, E \, (R_1+R_2)^2} \tag{5.2.2}
 \end{align} \end{align}
-and that this dependence is plotted in the lower panel of\cref{fig:HardBallScatteringAngle}. +and that this dependence is plotted in the lower panel of\cref{fig:HardBallScatteringAngle}.\\ 
-  -  What happens when $L^2 > 2\mu \, E \, (R_1+R_2)^2$? \\ + 
-Which angle $\theta$ will one observe in that case? +**d)** What happens when $L^2 > 2\mu \, E \, (R_1+R_2)^2$? Which angle $\theta$ will one observe in that case?\\ 
-  -  **$\star$** Show that\cref{eq:HardBallScatteringAngle,eq:CoulombScatteringAngle} agree when one identifies + 
-the length scale $R_1+R_2$ of the hard-ball system with the distance $R_{\text{eff}}$ of symmetry point of the cone section from the origin, +**e)** :!: Show that [[#eq_HardBallScatteringAngle |Equation 5.2.2]] and [[book:chap5:5.1_motivation_and_outline #eq_CoulombScatteringAngle |Equation 5.1.1]] agree when one identifies 
-i.e., with the mean value of the two intersection points with the $\hat x$-axis+the length scale $R_1+R_2$ of the hard-ball system with the distance $R_{\text{eff}}$ of symmetry point of the cone section from the origin, i.e., with the mean value of the two intersection points with the $\hat x$-axis
 \begin{align*}  \begin{align*} 
       R_{\text{eff}}       R_{\text{eff}}
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 ----  ---- 
 +
 +<WRAP 120pt right>
 +{{02_billiard_A1.png}}
 +</WRAP>
  
 <wrap #quest_Conservation-05 > Problem 5.4: </wrap> <wrap #quest_Conservation-05 > Problem 5.4: </wrap>
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 \\ \\
 The sketch to the right shows a billiard table. The sketch to the right shows a billiard table.
-The white ball should be kicked (i.e. set into motion with velocity $\mathbf v$), +The white ball should be kicked (i.e. set into motion with velocity $\mathbf v$), and hit the black ball such that it ends up in pocket to the top right. What is tricky about the sketched track?What might be a better alternative?
-and hit the black ball such that it ends up in pocket to the top right. +
-\\ +
-What is tricky about the sketched track? +
-\\ +
-What might be a better alternative?+
  
-<WRAP right> 
-{{./Sketch/02_billiard_A1.png}} 
-</WRAP> 
  
  
 ~~DISCUSSION~~  ~~DISCUSSION~~ 
  
book/chap5/5.2_collisions_of_particles.1641263223.txt.gz · Last modified: 2022/01/04 03:27 by abril