Kinetic and Potential Energy Equal with

[theyellowking]

Question: "A 2kg ball is thrown upwards with a speed of 40m/s. At what height will the ball be, when its KE is equal to its PE?"

So my initial thinking was to set up as: Energy (initial) = Energy (final)
0.5mvi^2 + 0 = 0.5mvf^2 + mgh
vi^2 = vf^2 + 20h
1600 = vf^2 + 20h

It was here that I got stuck, and when I looked at the back for the answers, it was as follows:
"All its kinetic energy changes to potential energy; potential energy is proportional to height, thus at halfway the energies are equal. Find the maximum height from v = sqrt(2gh) and divide by 2"

My problem is the author's conclusion of "thus at halfway the energies are equal." What made him/her conclude this fact? Why is the KE equal to PE at the halfway mark? And is there a way from my set-up to get the answer?

The answer is 40m, by the way.

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[JMMTB]

Initially the total E is the KE that we give the ball KE=(1/2)mv^2=(1/2)(2)(40)^2= 1600 Joules

If we want to find when KE=PE then it must be when PE=(1/2)E Total or KEinital because KE +PE = E tot and if KE = PE. 2PE=E tot

PE=1600/2=800=mgh

h=800/(mg)=800J/(2kg*10m/s^2)=40m

 

[DrknoSDN]

Just figure what max PE is and cut it in half. Thats half PE so by extention KE must be equal to PE.

Max height is 20m/s (avg) * 4 seconds. = 80meters
Height = 40m.