Let's go through your derivation step-by-step to identify where the mistake occurred.
1. You correctly started with the definition of the average velocity as t^(-1) (x - x0).
2. Then you wrote x = (average velocity) * t + x0, which is correct.
3. Next, you used the formula for average velocity when constant acceleration is assumed, v = 2^(-1) (v0 + v). However, you made a mistake in substituting this into the previous equation. It should be x = (average velocity) * t + x0 = 2^(-1) (v0 + v) * t + x0 (not 2^(-1) (v0 + v) * t + x0).
4. You correctly wrote the definition of acceleration as a = t^(-1) (v - v0).
5. Then you made a mistake when substituting t = a^(-1) (v - v0) into x = 2^(-1) (v0 + v) * t + x0. It should be x = 2^(-1) (v0 + v) * t + x0 = 2^(-1) (v0 + v) * (a^(-1) (v - v0)) + x0 (not 2^(-1) (v0 + v) * t + x0).
6. When simplifying the expression, you made an error. It should be x = 2^(-1) (v0 + v)(a^(-1))(v - v0) + x0.
7. You mistakenly wrote (x = (a2)^(-1)(v^2 - v0^2) + x0)2a instead of simplifying the expression as (2a)x = (v^2 - v0^2) + x0.
8. Continuing with the simplification, you incorrectly wrote (2a)x - x0 + v0^2 = (v^2 - v0^2) + v0^2. It should be (2a)x - x0 + v0^2 = (v^2 - v0^2) - v0^2.
9. Finally, you made an error in combining the terms, resulting in v^2 = v0^2 + (2a)x - x0 + v0^2, instead of v^2 = v0^2 + 2a(x - x0).
To correct the mistake, you need to fix step 5 and continue the simplification correctly. Here's the corrected derivation:
1. Definition of average velocity: v = t^(-1) (x - x0).
2. Rearrange to solve for x: x = vt + x0.
3. Formula for average velocity with constant acceleration: v = 2^(-1) (v0 + v).
4. Substituting into x = vt + x0: x = 2^(-1) (v0 + v) t + x0.
5. Definition of acceleration: a = t^(-1) (v - v0).
6. Substituting t = a^(-1) (v - v0) into x = 2^(-1) (v0 + v) t + x0:
x = 2^(-1) (v0 + v) (a^(-1) (v - v0)) + x0.
7. Simplifying the expression: x = 2^(-1) (v0 + v) (a^(-1)) (v - v0) + x0.
8. Multiply through by 2a: 2ax = v0v - v0^2 + v^2 - v0v + 2x0a.
9. Combine like terms: 2ax - 2x0a = v^2 - v0^2.
10. Rearrange the equation: v^2 = v0^2 + 2a(x - x0).
So the correct derivation is:
v^2 = v0^2 + 2a(x - x0)