A $1\mathrm{.}50-$kg object slides to the right on a surface having a coefficient of kinetic friction $0\mathrm{.}250($Figure a$).$ The object has a speed of vi $=2\mathrm{.}70$ m$/$s when it makes contact with a light spring $($Figure b$)$ that has a force constant of $50\mathrm{.}0$ N$/$m$.$ The object comes to rest after the spring has been compressed a distance d $($Figure c$).$ The object is then forced toward the left by the spring $($Figure d$)$ and continues to move in that direction beyond the spring's unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring $($Figure e$).$

The right end of a horizontal spring labeled k is attached to a wall. Five images show five configurations as a block labeled m approaches, compresses, and then moves away from the spring.

In figure a$,$ the block is to the left of the spring, and an arrow above the block points to the right.

In figure b$,$ the block is just touching the uncompressed spring, and an arrow labeled vector vi above the block points to the right.

In figure c$,$ the block has compressed the spring by a distance d$,$ and a label indicates vector vf $=0.$

In figure d$,$ the block is just touching the uncompressed spring, and an arrow labeled vector v above the block points to the left.

In figure e$,$ the block is a distance D away from the spring, and a label indicates vector v $=0.$

$($a$)$

Find the distance of compression d $($in m$).$

m

$($b$)$

Find the speed v $($in m$/$s$)$ at the unstretched position when the object is moving to the left $($Figure d$).$

m$/$s

$($c$)$

Find the distance D $($in m$)$ where the object comes to rest.

m

$($d$)$

What If$?$ If the object becomes attached securely to the end of the spring when it makes contact, what is the new value of the distance D $($in m$)$ at which the object will come to rest after moving to the left?

m