Problem $7$: A block of mass $3\mathrm{.}4kg$ is sitting on a frictionless ramp with a spring at the bottom that has a spring constant of $550\frac{N}{m}($refer to the figure$).$ The angle of the ramp with respect to the horizontal is $11.$

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Part $($a$)$ The block, starting from rest, slides down the ramp a distance $46cm$ before hitting the spring. How far, in centimeters, is the spring compressed as the block comes to momentary rest? Numeric : A numeric value is expected and not an expression.

$x=$

Part $($b$)$ After the block comes to rest, the spring pushes the block back up the ramp. How fast, in meters per second, is the block moving right after it comes off the spring?

Numeric : A numeric value is expected and not an expression.

$v=$

Part $($c$)$ What is the change of the gravitational potential energy, in joules, between the original position of the block at the top of the ramp and the position of the block when the spring is fully compressed?

Numeric : A numeric value is expected and not an expression.

$U_g=$