Let $\tilde{t}=\frac{t}{t_{}s}$ and $\tilde{y}(\tilde{t})=\frac{y(t)}{y_{s}}$ denote nondimensional time and height respectively with some scales $t_{s}$ and $y_{s}$. By chain rule $ \begin{aligned} \frac{d}{d t} y(t) & =\frac{d}{d t}\left(y_s \tilde{y}\left(\frac{t}{t_s}\right)\right)=y_s \tilde{y}^{\prime}\left(\frac{t}{t_s}\right) \frac{1}{t_s}=\frac{y_s}{t_s} \tilde{y}^{\prime}(\tilde{t}), \\ \frac{d^2}{d t^2} y(t) & =\frac{d}{d t}\left(y_s \tilde{y}^{\prime}\left(\frac{t}{t_s}\right) \frac{1}{t_s}\right)=y_s \tilde{y}^{\prime \prime}\left(\frac{t}{t_s}\right) \frac{1}{t_s^2}=\frac{y_s}{t_s^2} \tilde{y}^{\prime \prime}(\tilde{t}) . \end{aligned} $ The IVP becomes $ \begin{align} \frac{y_s}{t_s^2} \tilde{y}^{\prime \prime}(\tilde{t})&=-\frac{g}{\left(1+\tilde{y}(\tilde{t}) \frac{y_s}{R}\right)^2}, \quad \tilde{t}>0, \quad \tilde{y}(0)=0, \quad \frac{y_s}{t_s} \tilde{y}^{\prime}(\tilde{t})=v \\ \iff \frac{y_s}{gt_s^2} \tilde{y}^{\prime \prime}(\tilde{t})&=-\frac{g}{\left(1+\tilde{y}(\tilde{t}) \frac{y_s}{R}\right)^2}, \quad \tilde{t}>0, \quad \tilde{y}(0)=0, \quad \tilde{y}^{\prime}(\tilde{t})=\frac{t_{s}}{y_{s}}v \end{align} $ We note that the dimensions of the parameters are $[v] = L/T , [g] = L/T^2, [R] = L.$ Therefore, the parameters in the non-dimensional equation $\frac{y_{s}}{gt_{s}^{2}}, \frac{y_{s}}{R}, \frac{t_{s}v}{y_{s}}$are indeed non-dimensional. ...