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course PHY 231
9/26 9:33 pm
`qx025. (univ; gen invited) Can you interpret the expressions for v_mid_x and v_mid_t to answer at least some of the open questions associated with the ordering of v0, vf, `dv, v_mid_x, v_mid_t and v_ave? Can you develop expressions that can be interpreted in order to answer the remaining questions?
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Well I know now that v_mid_x will always be greater than v_mid_t since it takes more time to get to v_mid_x than t/2. All of my equations have vf growing larger as v0 grows larger, which is expected and well known. I have suspected all along that v_mid_t = v_Ave and here are my equations that prove it: v_Ave = `dx/t = (v0*t+1/2*a*t^2)/t = v0+1/2*a*t and, since mid_t = t/2 I have: v_mid_t = v0+ a(t/2)= v0 +1/2*a*t which are clearly the same. Another idea that had early on is that as v0 grows larger `dv will grow smaller and eventually be at the bottom of the list and here is proof of that: `dv = vf-v0, vf= sqrt(v0^2+2*a*`dx) then substitution gives: `dv = (sqrt(v0^2+2*a*`dx))-v0. This shows that for different trials, with the same a and `dx but increasing v0, the second half of the square root becomes negligible as compared to v0 and so we get something very close to sqrt(v0^2)-v0 which is 0, but `dv will never be 0 if a and/or `dx are always > 0. So the only real change in my first assertion about the ordering of these is that v_mid_x and v_Ave/v_mid_t are not equal but v_Ave and v_mid_t still are. So as v0 increases from 0 the only quantity that moves in the ordering is `dv which moves down the ranks from being next to vf at the top, when v0 = 0, to less than v0 at the bottom when v0 gets very large, and the rest are v0 < v_Ave = v_mid_t < vf.
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Just to add to this a little. I know, but did not reiterate above, that when v0=0 then `dv = vf. More interestingly though is that I saw that there is a v0 such that `dv = v0. This will occur when v0 = sqrt(2/3*a*`dx) based on the equation I used above: `dv = sqrt(v0^2+2*a*`dx)-v0. This is surprising and something I had not anticipated until I thought about `dvv0 to `dv
The coasting idea makes sense now. And all of these math characters I use are available anywhere you can type on a mac so I know how to read it. In fact I'm typing this right into the submit field in firefox and can use: ∫ sqrt π ø Ø Ω ≈ ≠ ± ≤ ≥ `d ∑ ƒ ∂ and many many others. Really useful in these QAs.
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Great.
I changed your characters into text notation. The key you included can be used to translate the characters. Very easy to work that into my program so it's done automatically. Can't do it right now; midnight deadline for posting today is only a few minutes away, but will do it soon.