Kinematic Equations and Free Fall

The kinematic equations (constant acceleration only)

Equation	Missing variable
v = v₀ + at	Δx
Δx = v₀t + (1/2)at²	v
v² = v₀² + 2aΔx	t
Δx = (1/2)(v₀ + v)t	a

Method

1. Choose a positive direction.
2. List the knowns (three are needed) and the unknown.
3. Pick the equation that leaves out the quantity you do not know and do not need.
4. Check units and signs.

Example. A car accelerates from 10 m/s at 2 m/s² for 5 s.

v = 10 + 2(5) = 20 m/s; Δx = 10(5) + (1/2)(2)(25) = 75 m.

Example (braking). A car at 30 m/s stops in 45 m. v² = v₀² + 2aΔx: 0 = 900 + 90a, so a = −10 m/s².

Free fall

Near Earth's surface, ignoring air resistance, every object accelerates downward at g ≈ 9.8 m/s² (often rounded to 10 m/s²).

• At the top of its path, a thrown object has v = 0 but its acceleration is still g downward.
• Time up equals time down to the same height.

Example (g = 10 m/s²). A ball thrown up at 20 m/s: time to the top = 20/10 = 2 s; maximum height = 20²/(2 × 10) = 20 m.

Common mistakes

• Using the equations when acceleration is not constant.
• Mixing signs: if up is positive, g is −10 m/s².
• Thinking acceleration is zero at the top.

Quick summary

• Four equations, each missing one variable.
• Free fall: a = g downward the whole time.
• At the top, v = 0, a = g.