Lesson 2 · 14 min
Constant acceleration
Four equations. They only hold when a is constant. That is most of freshman physics.
The contract
If acceleration is constant — gravity near Earth, a brake with constant force, a piston in a simplified stroke — the derivatives integrate in closed form. You get four relations among x, v, a, and t. You never need all four at once; you pick the one that omits the unknown you do not have.
v = v₀ + at
Use when you have no displacement.
Δx = v₀t + ½at²
Use when you have no final velocity.
v² = v₀² + 2aΔx
Use when you have no time. The work–energy theorem will later explain why this looks like energy.
Worked example
A braking truck
- Given: v₀ = 25 m/s (90 km/h)
- Given: v = 0
- Given: a = −6.0 m/s² (anti-lock, roughly constant)
Find: Stopping distance
- No time given — use v² = v₀² + 2aΔx.
- 0 = (25)² + 2(−6)Δx
- 0 = 625 − 12 Δx
- Δx = 625 / 12
52 m (a city block is ~80 m — this truck needs most of one)
Check
You know v₀, v, and Δx, and you want a. Which equation is the direct path?