Deriving kinematic equation from acceleration definition via chain rule

Steps:

• Define constant acceleration: a = dv/dt.
• Apply chain rule: dt = ds/v, so a = v dv/ds.
• Rearrange: v dv = a ds.
• Integrate limits (u to v for velocity, 0 to s for displacement): ∫_u^v v dv = a ∫_0^s ds → (v² - u²)/2 = as → v² = u² + 2as.

Why C is correct:

• Derived directly from acceleration definition a = dv/dt using chain rule and integration over displacement, avoiding time explicitly.

Why the others are wrong:

• A: Requires integrating velocity over time to get displacement.
• B: Uses average acceleration over time interval Δt.
• D: Identical to C, but not the specified unique option.

Final answer: C