  We show that the classical Kac's random walk on $(n-1)$-sphere $S^{n-1}$
starting from the point mass at $e_1$ mixes in $\mathcal{O}(n^5(\log n)^3)$
steps in total variation distance. The main argument uses a truncation of the
running density after a burn-in period, followed by $\mathcal{L}^2$ convergence
using the spectral gap information derived by other authors. This improves upon
a previous bound by Diaconis and Saloff-Coste of order $\mathcal {O}(n^{2n})$.
