  We investigate stochastic resetting in coupled systems involving two degrees
of freedom, where only one variable is reset. The resetting variable, which we
think of as hidden, indirectly affects the remaining observable variable
through correlations. We derive the Fourier-Laplace transform of the observable
variable's propagator and provide a recursive relation for all the moments,
facilitating a comprehensive examination of the process. We apply this
framework to inertial transport processes where we observe particle position
while the velocity is hidden and is being reset at a constant rate. We show
that velocity resetting results in a linearly growing spatial mean squared
displacement at late times, independently of reset-free dynamics, due to
resetting-induced tempering of velocity correlations. General expressions for
the effective diffusion and drift coefficients are derived as function of
resetting rate. Non-trivial dependence on the rate may appear due to multiple
timescales and crossovers in the reset-free dynamics. An extension that
incorporates refractory periods after each reset is considered, where the
post-resetting pauses can lead to anomalous diffusive behavior. Our results are
of relevance to a wide range of systems, including inertial transport where
mechanical momentum is lost in collisions with the environment, or the behavior
of living organisms where stop-and-go locomotion with inertia is ubiquitous.
Numerical simulations for underdamped Brownian motion and the random
acceleration process confirm our findings.
