We use the Fourier based Gabetta–Toscani–Wennberg metric $$d_2$$ d 2 to study the rate of convergence to equilibrium for the Kac model in 1 dimension. We take the initial velocity distribution of the particles to be a Borel probability measure $$\mu $$ μ on $$\mathbb {R}^n$$ R n that is symmetric in all its variables, has mean $$\vec {0}$$ 0 → and finite second moment. Let $$\mu _t(dv)$$ μ t ( d v ) denote the Kac-evolved distribution at time t, and let $$R_\mu $$ R μ be the angular average of $$\mu $$ μ . We give an upper bound to $$d_2(\mu _t, R_\mu )$$ d 2 ( μ t , R μ ) of the form $$\min \left\{ B e^{-\frac{4 \lambda _1}{n+3}t}, d_2(\mu ,R_\mu )\right\} ,$$ min B e - 4 λ 1 n + 3 t , d 2 ( μ , R μ ) , where $$\lambda _1 = \frac{n+2}{2(n-1)}$$ λ 1 = n + 2 2 ( n - 1 ) is the gap of the Kac model in $$L^2$$ L 2 and B depends only on the second moment of $$\mu $$ μ . We also construct a family of Schwartz probability densities $$\{f_0^{(n)}: \mathbb {R}^n\rightarrow \mathbb {R}\}$$ { f 0 ( n ) : R n → R } with finite second moments that shows practically no decrease in $$d_2(f_0(t), R_{f_0})$$ d 2 ( f 0 ( t ) , R f 0 ) for time at least $$\frac{1}{2\lambda }$$ 1 2 λ with $$\lambda $$ λ the rate of the Kac operator. We also present a propagation of chaos result for the partially thermostated Kac model in Tossounian and Vaidyanathan (J Math Phys 56(8):083301, 2015).