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Box2D is an open source C++ physics engine for simulating collisions and other 2D motions of rigid bodies which can be easily integrated with Qt. The main developer of Box2D is Erin Catto and it is distributed under the zlib license.
Box2D is widely used by games including Angry Birds and Angry Monsters.
For integrating Box2D in QT, just open the CMakeLists.txt in the source code.