Deriving Formula #5 - V2² = V1² + 2ad

[j1249]

Derive formula #5 using any of the other formulas

1. Vav= d/t
2. Aav= V2-V2/t
3. d= (V1+V2/2) t
4. d= V1t + 1/2 a t squared
5. V2 squared = V1 squared + 2ad


*note all the Vav, Aav, a and d have a vector sign ontop and d and t always have the triangle representing change in

 

[Hootenanny]

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[nlightner]

*

To derive formula #5, we can start with formula #4, which states that d = V1t + 1/2at². We can rearrange this equation to solve for V2 by subtracting V1t and dividing by 2 on both sides:
d - V1t = 1/2at²
2(d - V1t) = at²
2d - 2V1t = at²
2d = at² + 2V1t
Dividing both sides by t, we get:
2d/t = at + 2V1
Next, we can substitute formula #2 (Aav = (V2-V1)/t) into the equation:
2d/t = (V2-V1)t + 2V1
Multiplying both sides by t, we get:
2d = (V2-V1)t² + 2V1t
Finally, we can rearrange this equation to solve for V2:
2d - 2V1t = (V2-V1)t²
2d - 2V1t = V2t² - V1t²
2d - 2V1t = V2(t² - t² + 2at²)
2d - 2V1t = V2(2at²)
Dividing both sides by 2at², we get:
(2d - 2V1t)/2at² = V2
Simplifying, we get:
d - V1t = V2
Finally, we can square both sides to get formula #5:
V2² = (d - V1t)² = d² - 2dV1t + V1²t²
Substituting in formula #1 (Vav = d/t) and rearranging, we get:
V2² = Vav²t² - 2VavV1t + V1²t²
Simplifying, we get:
V2² = Vav²t² - 2VavV1t + V1²t²
Rearranging, we get:
V2² = V1² + 2VavV1t + Vav²t²
Substituting in formula #3 (d = (V1+V2)/2t), we get: