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| 両方とも前のリビジョン前のリビジョン次のリビジョン | 前のリビジョン | ||
| research:memos:kinematics:non-relativistic_kinematics [2017/10/26 22:50] – [Formulae for Non-relativistic Kinematics] kobayash | research:memos:kinematics:non-relativistic_kinematics [2020/07/29 17:10] (現在) – [Formulae for Non-relativistic Kinematics] kobayash | ||
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| 行 65: | 行 65: | ||
| \tan\theta_3 = \frac{\sin\theta_3^*}{1+\cos\theta_3^*}\\ | \tan\theta_3 = \frac{\sin\theta_3^*}{1+\cos\theta_3^*}\\ | ||
| \end{align}| | \end{align}| | ||
| - | | 8. lab to c.m. angle transformation ($\theta_\mathrm{cm} \rightarrow \theta_\mathrm{lab}$) |\begin{align} | + | | 8. lab to c.m. angle transformation ($\theta_\mathrm{lab} \rightarrow \theta_\mathrm{cm}$) |\begin{align} |
| - | \cos\theta_3^*=\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\pm\sqrt{\left(\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\right)^2-\frac{\delta_{23}^{*2}\tan^2\theta_3-1}{\tan^2\theta_3+1}} | + | \cos\theta_3^*=-\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\pm\sqrt{\left(\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\right)^2-\frac{\delta_{23}^{*2}\tan^2\theta_3-1}{\tan^2\theta_3+1}} |
| \end{align} Another solution (not written in the document) \begin{align} | \end{align} Another solution (not written in the document) \begin{align} | ||
| \tan\theta_{\rm cm} = \frac{\sin\theta_{\rm lab}}{\left(\cos\theta_{\rm lab}-v_{\rm cm}/ | \tan\theta_{\rm cm} = \frac{\sin\theta_{\rm lab}}{\left(\cos\theta_{\rm lab}-v_{\rm cm}/ | ||
| 行 359: | 行 359: | ||
| is | is | ||
| \begin{align} | \begin{align} | ||
| - | x=\frac{b}{a}\pm\sqrt{\left(\frac{b}{a}\right)^2-\frac{c}{a}}. | + | x=-\frac{b}{a}\pm\sqrt{\left(\frac{b}{a}\right)^2-\frac{c}{a}}. |
| \end{align} | \end{align} | ||
| Therefore, | Therefore, | ||
| \begin{align} | \begin{align} | ||
| - | \cos\theta_3^*=\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\pm\sqrt{\left(\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\right)^2-\frac{\delta_{23}^{*2}\tan^2\theta_3-1}{\tan^2\theta_3+1}}. | + | \cos\theta_3^*=-\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\pm\sqrt{\left(\frac{\delta_{23}^*\tan^2\theta_3}{\tan^2\theta_3+1}\right)^2-\frac{\delta_{23}^{*2}\tan^2\theta_3-1}{\tan^2\theta_3+1}}. |
| \end{align} | \end{align} | ||
