Fast dynamic movement is often a problem for Cloth Solvers/Simulators. Extremely fast dynamic movement such as this requires a much more precise simulation: this means that you have to increase the Simulation "Steps" Per Frame parameter to the extreme. This will slow down your simulation time significantly (hours instead of minutes). However, you will get a more precise/correct simulation as a result.
If you cannot wait the numerous hours it takes to simulate a cloth simulation with high steps per frame, then consider reducing the subdivision on your 2D mesh/plane. In this way, it will simulate faster due to less mass/particles to simulate. Note: this will help you gain simulation speed, as you also sacrifice realism.
Finally, If you've tried the highest "Steps" possible and it still isn't giving you sufficient correctness/stability for the simulation. Then you might consider using a Cloth Solver/Simulator that uses Implicit Solvers (Witkin, Baraff 1998) as opposed to Explicit Solver (I suspect that the Blender's cloth solver is using an Explicit Solver).
Reference for Witkin, Barraf 1998: http://www.cs.cmu.edu/~baraff/papers/sig98.pdf