Why is the Kalman gain constant when position and velocity change?
The Kalman Gain $K_k$ for a discrete, linear system is computed using the state covariance matrix $P_k$, measurement matrix $H_k$ and measurement noise covariance matrix $R_k$:

$$
K_k = P_kH_k^T(H_kP_kH_k^T+R_k)^{-1}
$$

If the measurement matrix and the measurement noise covariance matrix are constant and the state covariance matrix converges to a steady-state value $P_\infty$, the Kalman Gain also converges to a steady-state value $K_\infty$, regardless of actual measurements or state.