Physic I

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1)  The girl jumping on the skateboard is an inelastic collision, and in all collisions, energy must be conserved.  In an inelastic collision the combined energy of the colliding objects to before the collision, must equal the energy of the combined objects after the collision. 

Before the collision, the girl had Kinetic Energy `` And the skateboard, had zero (it was stationary or v=0 m/s).  So the two objects had 137.2 J of Kinetic energy. 

After the collision the girl and skateboard still have 137.2 but a velocity of 2.6 m/s and an unknown mass.  We can find the mass of the skateboard by solving the equation `` simply `` so `` kg combined between the girl and the skateboard.  We subtract the girl's 35 kg, and we are left with `` The skateboard's mass is 5.59kg

2) The three points form a triangle.  And the center of mass of a triangular object of uniform thickness is the centroid of the triangle.  The centroid of a triangle is where the medians of the triangle meet.  The medians are lines drawn from each vertex to the midpoint of the opposite side. 

We label your points A(2,-3), B(-4,2) and C(3,3).  The midpoint of `` is d, coordinates are `` or ``
The midpoint of `` is e, coordinates are `` or ``
The midpoint of `bar(CA)` is f, coordinates are `(3+2)/2, (3+ -3)/2` or `(2 1/2, 0)`
From these we can calculate the slope of median `bar(Cd)` to be `(-1/2-3)/(-1-3)=(-3 1/2)/-4=7/8` we place that in point intercept form `y-3=7/8(x-3)` and then simplify to slope intercept:`y=7/8x+3/8`

The slope of median `bar(Ae)` to be `(2 1/2- -3)/(-1/2-2)=(5 1/2)/(-2 1/2)=-2 1/5` we place that in point-intercept form `y+3=-2 1/2(x-2)` and simplify to slope intercept `y=-2 1/2x+2`
and the slope of Median `bar(Bf)` is `(0-2)/(2 1/2+4)=(-2)/(6 1/2)=-4/13`` ` we place that in point-intercept form `y-2=-4/13(x+4)` and simplify to slope intercept `y=-4/13x+10/13`
We solve the three equations for the point they all have in common, the centroid.  `-2 1/2x+2=7/8x+3/8` we add 2.5x to each side to combine like terms and get `2=27/8x+3/8` we subtract 3/8 from each side to isolate the variable.  `13/8=27/8x` we divide each side by 27/8 and we get `x=bar.481` we plug that into our third equation and `y=-4/13(bar.481)+10/13~~.621` We round these two to two significant digits and we have coordinates (.48,.62)

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