Posted by James on Friday, May 18, 2012 at 4:43pm.
Maybe it is a reading of the question. If the max height is y, midpoint is y/2
If that is so, then the 9.8 should be -9.8, and the equation is dead on correct.
The motion to the highest point (height “H”)
v = vₒ - g•t,
v=0, t = vₒ/g.
H = vₒ•t - g•t²/2 = vₒ• vₒ/g –(g/2) •(vₒ/g)² = vₒ²/2•g.
Velocity at the midpoint (height “h“)
v =vₒ - g•t1,
t1 =(vₒ- v)/g.
h = vₒ•t1 - g•t1²/2 =
=vₒ•(vₒ- v)/g - g•(vₒ- v)²/2•g² =
= [2•vₒ•(vₒ- v) -(vₒ- v)²]/ 2•g =
={2vₒ² - 2vₒ•v - vₒ² +2 vₒ•v - v²}/2•g =
= (vₒ² - v²)/2•g.
h=H/2 = vₒ²/2•2•g.
(vₒ² - v²)/2•g = vₒ²/2•2•g.
vₒ² = 2•v²,
v =vₒ/√2 = vₒ/1.41 =0.707•vₒ
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