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If someone who is at rest initially, jumps off a frictionless cart in the horizontal direction, there are no external forces, only internal. However, the person and the cart gain kinetic energy. As a result, energy is not conserved. But why does this happen, if there are no external forces? (I know the COM of the system is not moving, but the elements of the systems are moving, so extra energy was introduced)

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The kinetic energy was converted from the chemical potential energy in the person's muscles which caused the motion. Kinetic energy is not conserved, total energy is.

Think about a bomb going off. It starts as chemical potential energy, which then gets very suddenly released, partially as kinetic energy of all the shrapnel, and mainly as heat

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