Physics

To solve this problem, we analyze the system step by step:

 

Conservation of momentum to find the final velocity:

The initial momentum of the system is given by the momentum of the wooden block only since the cart is stationary. The momentum of the block is , where and . So .

 

After the block slides onto the cart, the two objects move together. The combined mass , where . Thus .

 

By conservation of momentum , where is the final velocity of the combined objects. So .

 

Find the acceleration of the block and the cart:

The frictional force acting on the block is , where the normal force .

The frictional force , with . So .

According to Newton's second law, the acceleration of the block relative to the cart is . Substitute and . So .

 

By Newton's Third Law, the force of friction acting on the cart is equal and opposite to the force on the block:

The acceleration of the cart is:

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