Consider a target moving with a constant velocity on a unit-circumference circle, starting from an arbitrary location. To acquire the target, any region of the circle can be probed for its presence, but the associated measurement noise increases with the size of the probed region. We are interested in the expected time required to find the target to within some given resolution and error probability. For a known velocity, we characterize the optimal tradeoff between time and resolution (i.e., maximal rate), and show that in contrast to the case of constant measurement noise, measurement dependent noise incurs a multiplicative gap between adaptive search and non-adaptive search. Moreover, our adaptive scheme attains the optimal rate-reliability tradeoff. We further show that for optimal non-adaptive search, accounting for an unknown velocity incurs a factor of two in rate.