A The Higgs Field, constant velocity, and the acceleration of particles

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Summary
Does the Higgs field affect constant moving particles or accelerating particles
I need some clarification on how the Higgs field work. The popular science explanation explains the effect on moving particles the way an object would be impacted by moving through a medium of maple syrup. I understand this is a very bad, inaccurate analogy. What I want to understand is does the Higgs field effect constant moving particles or accelerating particles?
 

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