  In this work, we study the effect of a moving detector on a discrete time one
dimensional Quantum Random Walk where the movement is realized in the form of
hopping/shifts. The occupation probability $f(x,t;n,s)$ is estimated as the
number of detection $n$ and amount of shift $s$ vary. It is seen that the
occupation probability at the initial position $x_D$ of the detector is
enhanced when $n$ is small which is a quantum mechanical effect but decreases
when $n$ is large. The ratio of occupation probabilities of our walk to that of
an Infinite walk shows a scaling behavior of $\frac{x_D^2}{n^2}$. It shows a
definite scaling behavior with amount of shifts $s$ also. The limiting
behaviors of the walk are observed when $x_D$ is large, $n$ is large and $s$ is
large and the walker for these cases approach the Infinite Walk, The Semi
Infinite Walk and the Quenched Quantum Walk respectively.
