That`s just wrong. If e = mc^2 THAN m is NOT accelerated e...
[c^2] = [(v^2/t^2)] != [(v/t^2)] = [a]
We can also choose natural units where c==1, which leads directly to e==m
It's why I said "broadly what the equation shows." It's hard to tell at rest, but it is still technically including the momentum.
"does water have ice?" same energy