Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ with respect to $I$ and let $\mathcal{R}=A[It,t^{-1}]$ be the extended Rees ring of $A$ with respect to $I$. Notice $t^{-1}$ is a non-zerononzero divisor on $\mathcal{R}$ and $\mathcal{R}/t^{-1}\mathcal{R}=G$. So, we have Bockstein operators ${\beta }^i:H_{G_{+}}^i(-1)\to H_{G_{+}}^{i+1}(G)$ for $i\ge 0$. Since ${\beta }^{i+1}(+1)\circ {\beta }^i=0$$, we have Bockstein cohomology modules $B{H}^i(G)$ for $i=0,\dots ,d$. In this paper, we show that certain natural conditions on $I$ implies vanishing of some Bockstein cohomology modules.