To navigate in an environment safely and autonomously, robots must accurately estimate where obstacles are and how they move. Instead of using expensive traditional 3D sensors, we explore the use of a much cheaper, faster, and higher resolution alternative: programmable light curtains. Light curtains are a controllable depth sensor that sense only along a surface that the user selects. We adapt a probabilistic method based on particle filters and occupancy grids to explicitly estimate the position and velocity of 3D points in the scene using partial measurements made by light curtains. The central challenge is to decide where to place the light curtain to accurately perform this task. We propose multiple curtain placement strategies guided by maximizing information gain and verifying predicted object locations. Then, we combine these strategies using an online learning framework. We propose a novel self-supervised reward function that evaluates the accuracy of current velocity estimates using future light curtain placements. We use a multi-armed bandit framework to intelligently switch between placement policies in real time, outperforming fixed policies. We develop a full-stack navigation system that uses position and velocity estimates from light curtains for downstream tasks such as localization, mapping, path-planning, and obstacle avoidance. This work paves the way for controllable light curtains to accurately, efficiently, and purposefully perceive and navigate complex and dynamic environments. Project website: https://siddancha.github.io/projects/active-velocity-estimation/