A block with mass m = 12 kg rests on a frictionless table and is accelerated by a spring with spring constant k = 4822 N/m after being compressed a distance x1 = 0

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A block with mass m = 12 kg rests on a frictionless table and is accelerated by a spring with spring constant k = 4822 N/m after being compressed a distance x₁ = 0.408 m from the spring’s unstretched length. The floor is frictionless except for a rough patch a distance d = 2 m long. For this rough path, the coefficient of friction is μ_k = 0.44.

How much work is done by the spring as it accelerates the block?

What is the speed of the block right after it leaves the spring?

How much work is done by friction as the block crosses the rough spot?

What is the speed of the block after it passes the rough spot?

Instead, the spring is only compressed a distance x₂ = 0.118 m before being released. How far into the rough path does the block slide before coming to rest?

What distance does the spring need to be compressed so that the block will just barely make it past the rough patch when released?

If the spring was compressed three times farther and then the block is released, the work is done on the block by the spring as it accelerates the block is: the same, three times greater, three times less, nine times greater, or nine times less ???