Problem $11$: $(8\backslash \%$ of Assignment Value$)$

Laboratory students perform an experiment utilizing a wooden plank propped with a crate at an angle of $\backslash (13\mathrm{.}3^\{\backslash$circ$\}\backslash )$ above the floor. As pictured in the drawing below, a first block, with mass $\backslash (\backslash$mathrm$\{$M$\}=711\backslash$mathrm$\{$~g$\}\backslash ),$ is held at rest against a spring with a spring constant of $\backslash (11\mathrm{.}7\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ that is compressed by $4\mathrm{.}25$ cm $.$ When the block is released, it slides up the plank a distance of $\backslash (\backslash$mathrm$\{$d$\}=86\mathrm{.}9\backslash$mathrm$\{$~cm$\}\backslash )$ to where it collides with a second block of mass $\backslash (\backslash$mathrm$\{$m$\}=232\backslash$mathrm$\{$~g$\}\backslash ).$ Due to some embedded magnets, the blocks stick together, and they travel as a unit up the plank, eventually going past the end of the plank a distance $\backslash (\backslash$mathrm$\{$L$\}=121\mathrm{.}6\backslash$mathrm$\{$~cm$\}\backslash )$ from the original position of the first block. The coefficient of kinetic friction between the blocks and the plank is $0\mathrm{.}23.$ Ignore the finite size of the blocks.

$-$ Part $($a$)$

Calculate the speed, in meters per second, of the first block the instant before it collides with the second block?

$\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$A$\}\}=4\mathrm{.}670\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$

$\backslash (\backslash$checkmark $\backslash )$ Correct!

$-$ Part $($b$)\backslash (\backslash$boldsymbol$\{$v$\}\backslash )$

Calculate the speed, in meters per second, of the two blocks an instant after the collision?

$\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$B$\}\}=3\mathrm{.}520\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$

$\backslash (\backslash$checkmark $\backslash )$ Correct!

Part $($c$)$

Calculate the speed, in meters per second, of the two blocks at the instant they reach the end of the plank?

$\backslash (\backslash$mathrm$\{$v$\}\_\{\backslash$mathrm$\{$C$\}\}=\backslash )$

m$/$s