hi I need help on this proof
ok I'm tyring to prove that
v^2 = v^20 + 2a(x -x0)
where the zeros are subscripts
ok the first step my book says is
start out with
x = x0 + average velocity (t)
then plug in average velocity
x = x0 + (2^-1(v + v0))t
ok and the next line it says
solve for t
and the next formula it shows
is
t = a^-1(v - v0)
ok how did you go from that line to this one?
Please show me
what happened to the two?
were did the a come from
I don' get it
Thanks!
I do not blame you but I will try.
V^2 = Vo^2 + 2 a (X-Xo)
is where you want to go
ok
X = Xo + (V+Vo)t/2
V+Vo = 2 (X-Xo)/t
Now what you seem to be missing is
V = Vo + a t
which you are supposed to know, the definition of constant acceleration (change in velocity = acceleration times time)
so
t =(V-Vo)/a
then
(V+Vo) = 2(X-Xo) a /(V-Vo)
so then
V^2-Vo^2 = 2 (X-Xo) a
This is a really, really, klutzy way to do this !!!
To understand how the equation changed from x = x0 + (2^-1(v + v0))t to t = a^(-1)(v - v0), let's break it down step by step:
Starting with the equation x = x0 + (2^-1(v + v0))t:
1. Multiply both sides of the equation by 2 to eliminate the fraction:
2x = 2x0 + (v + v0)t
2. Rearrange the terms to isolate the "t" term on one side:
(v + v0)t = 2x - 2x0
3. Divide both sides by (v + v0) to solve for "t":
t = (2x - 2x0)/(v + v0)
Now, let's simplify the expression on the right side further:
4. Recognize that the difference between two positions (2x - 2x0) can be expressed as (x - x0) * 2:
t = (2(x - x0))/(v + v0)
5. Notice that the factor 2 in the numerator can be written as 2a, where "a" represents the average acceleration:
t = (2a(x - x0))/(v + v0)
6. Divide both sides by 2a to get "t" on its own:
t = (2a(x - x0))/(2a(v + v0))
7. Simplify the expression by canceling out the common factor of 2a:
t = (x - x0)/(v + v0)
8. Finally, invert the fraction to get the reciprocal:
t = (v + v0)/(x - x0)
At this point, we have obtained the equation t = (v + v0)/(x - x0). However, it is also common to represent acceleration as a = (v - v0)/(t), where "a" represents the acceleration, "v" represents the final velocity, "v0" represents the initial velocity, and "t" represents the time interval.
By comparing the equation t = (v + v0)/(x - x0) from step 8 with a = (v - v0)/(t), we can see that a^-1 (or 1/a) corresponds to (x - x0)/(v + v0).
Therefore, we can write:
t = a^(-1)(v - v0)
So, the formula changed from x = x0 + (2^-1(v + v0))t to t = a^(-1)(v - v0) because of the simplifications and rearrangements we made. The 2 in the original equation canceled out, and "a" represents the average acceleration.