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If a man starts running on a boat with an acceleration $a$ with respect to the boat, there is no external force that acts on the Boat+Man system (assuming friction due to water to be zero and neglecting air drag). But the velocity of system increases. What causes the increase in kinetic energy even if there is no external force?

I have not neglected friction between man and boat. But in boat+man system, its not an external force. Gravity does not do any work since displacement of boat is perpendicular to force due to gravity.

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It does not matter that there is no external force acting on the system. The kinetic energy comes from the man running on the boat. He is turning the chemical energy in his muscles into kinetic energy. If we have an isolated system (i.e. one with no external forces and where nothing leaves or enters the system), we require energy to be conserved within that system but not the conservation of kinetic energy, which I think is what you have assumed.

Let us take an analogy. You have a car stationary on a road with no other forces present except the friction between the road and the car. The car can start its engine and drive off, increasing the kinetic energy of the system. Here the kinetic energy comes from the car's fuel which is burnt in its engine. In this case the road will in fact move (like the boat moves) but since it is attached to the earth the motion of the road is far less noticeable than that of the boat.

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