$7\mathrm{.}18.$ A particle of mass $m_1$ with an initial kinetic energy $T_1$ makes an elastic collision with a particle of mass $m_2$ initially at rest. $m_1$ is deflected from its original direction with a kinetic $T_1^{\text{'}}$ through an angle ${}^{\text{'}}$ as in figure above. Letting $a=\frac{m_2}{m_1}$ and $=cos{}_1,$ show that the fractional kinetic energy lost by $m_1,\frac{T_1}{T_1}=\frac{T_1-T^{\text{'}}}{T_1},$ is given by

$\frac{T_1}{T_1}=\frac{2}{1+}-\frac{2}{1+{}^2}(\sqrt[\phantom{2}]{{}^2+{}^2-1})$

$7\mathrm{.}21$ A particle of mass $m_1$ scatters elastically from a particle of mass $m_2$ initially at rest as described in Problem $7\mathrm{.}19.$ Find the curve $r({}_1)$ such that the lime it takes the scattered particle to travel from the collision point to any point along the curve is a constant.