Problem #$8$: Impacts $($design a cranberry sorter$)$

In this problem you will be designing the key component of

a cranberry bounce sorter $($similar to the one you saw in the

video in class$)$ to sort ripe cranberries from the ones that are

over$-$ripe. To do this, we$$ll be making several simplifying

assumptions for the first draft of the design. We$$ll assume

the following:

$1)$ The cranberry size is negligible $($next step would

be to assume it$$s a sphere of a certain radius, not

true but a bit more accurate since the whole

cranberry will need to clear the wall after the bounce

not just the center of mass of the cranberry$)$

$2)$ The impact can be modeled as a frictionless

impact, most certainly not true but allows us to

simply the problem so we can actually make an

estimate here

$3)$ The wall has no thickness so once the point mass

passes over its height we don$$t have to worry about

it hitting the top of the wal $($maybe not a big deal

since the berry would likely bounce harmlessly off

the top but this will remove that possible issue$)$

$4)$ The coefficient of restitution for a ripe cranberry is

$0\mathrm{.}8$ and for an over$-$ripe cranberry it$$s $0\mathrm{.}2.$ NOTE

these are huge assumptions as I honestly just made

them up$.$ However, you could do some experiments

to try to get more accurate values for these. The design calculations will be the same.

$5)$ All cranberries have the same mass and are dropped, from rest, at a height above the ramp.

$6)$ All motion takes place in the $2$D vertical plane $($won$$t be true as the berries will bounce in weird directions, but

hopefully is close enough$).$

$7)$ Assume no air drag.

Question $1$: Determine $2$ expressions, one for the minimum and one for the maximum, height of the wall, $($in terms of

$)$ which will allow the ripe cranberries to pass over the wall and over$-$ripe cranberries to hit the wall and not

pass over it$.$ and $.$ A few things to note: the maximum height of the trajectory of

the cranberry will not necessarily occur at the location of the wall. The expressions that you develop should be able to be

used by someone to determine the max and min wall height if they have and can plug in values for $.$

Question $2$: After you have determined those two analytic expressions, plug in the following numeric values into your

expressions to calculate the maximum and minimum wall height $($ and $)$ for the following values $($m $=2[$g$],$ h $=$

$1[$m$],$ d $=0\mathrm{.}15[$m$],$ g $=9\mathrm{.}8[$m$/$s$^2$