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I know that for velocity conversions between ECI and ECEF there is an $\omega \times r_{ECI} $ term, such that the overall transformation is $v_{ECEF}=v_{ECI}-\omega \times r_{ECI}$. In my belief there shouldn’t be an extra term in the acceleration and forces conversions. For example, for a rocket at firing with a certain thrust force in the ECI frame $F_{T,ECI}$, the equivalent ecef force will be $F_{T,ECEF}=\mathbb{C}_{ECI}^{ECEF} F_{T,ECI}$, where $\mathbb{C}$ indicates the DCM from ECEF to ECI. The same would be true for acceleration, hence, for thrust acceleration vector: $w_{ECI}=\mathbb{C}_{ECI}^{ECEF} w_{ECEF}$ and vice versa if you wanted to go from ECI to ECEF. This should also be true for position. Can someone verify this?

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