Question 3.8: Making a change of variables from energy to speed, show that......
Making a change of variables from energy to speed, show that Eq. (2.40) becomes
p(v → v^′) = \Bigg \{\begin {matrix}\frac{2v^′}{ (1 – α) v²} , & v\sqrt{α} ≤ v^′ ≤ v \\ 0, & otherwise \end {matrix}
Since E\equiv{1/2}m\mathbf{v}^{2}\;\;{\mathrm{and}}\;\mathbf{v}={\sqrt{2E\,/\,m}}\;,\,\operatorname{we have}\;d E\,/\,d\mathbf{v}=m\mathbf{v}
p(\mathbf{v}\rightarrow\mathbf{v}^{\prime})=p(E\rightarrow E^{\prime})d E^{\prime}/d\mathbf{v}^{\prime} ={\frac{1}{(1-\alpha)E}}\,d E^{\prime}/\,d\mathbf{v}^{\prime}{=}{\frac{1}{(1-\alpha)1/2 m\mathbf{v}^{2}}}\,m\mathbf{v}^{\prime}
or p(\mathbf{v}\rightarrow\mathbf{v}^{\prime})={\frac{2\mathbf{v}^{\prime}}{(1-\alpha)\mathbf{v}^{2}}}
For the limits, when E^{\prime}=E , \mathbf{v}^{\prime}=\mathbf{v} and when E^{\prime}=\alpha E,\mathbf{v}^{\prime}={\sqrt{\alpha}}\mathbf{v}