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Derivation of Kinematic Equations: Step-by-Step Algebra Proofs

Understand exactly where the 4 kinematic equations come from. Clear, step-by-step algebraic derivations of the BIG 4 equations starting from basic definitions of velocity and acceleration.

Equation 1 Derivation All Four Equations

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Basic Definitions Derivation of Equation 1 Derivation of Equation 2 Derivation of Equation 3 Derivation of Equation 4 Summary Table FAQs

Basic Definitions Before Derivation

Before deriving the equations, recall these fundamental relations:

  • Velocity = change in displacement / time → v = Δx / t (average)
  • Acceleration = change in velocity / time → a = (v – v₀) / t
  • For constant acceleration, average velocity = (v₀ + v)/2

Derivation of Equation 1: v = v₀ + a t

Start with the definition of acceleration:

a = (v − v₀) / t

Multiply both sides by t:

a t = v − v₀

Rearrange to solve for final velocity v:

v = v₀ + a t

This is the first kinematic equation. It directly comes from the definition of acceleration.

Derivation of Equation 2: Δx = (v + v₀)/2 × t

For constant acceleration, the average velocity is the simple average of initial and final velocity:

v_avg = (v₀ + v) / 2

Displacement is average velocity multiplied by time:

Δx = [(v + v₀)/2] × t

This is Equation 2. It is especially useful when you know initial and final velocities but not acceleration.

Derivation of Equation 3: Δx = v₀ t + ½ a t²

We derive this by substituting Equation 1 into Equation 2.

From Equation 1: v = v₀ + a t

Plug into Equation 2:

Δx = [(v₀ + a t + v₀)/2] × t
Δx = [(2v₀ + a t)/2] × t
Δx = v₀ t + ½ a t²

Final result:

Δx = v₀ t + ½ a t²

Derivation of Equation 4: v² = v₀² + 2 a Δx

This equation is derived by eliminating time (t) from the previous equations.

Step 1: From Equation 1 → t = (v − v₀) / a

Step 2: Substitute this t into Equation 3:

Δx = v₀ [(v − v₀)/a] + ½ a [(v − v₀)/a]²

After simplifying algebraically (multiply both sides by 2a and rearrange), we get:

v² = v₀ ² + 2 a Δx

This equation is very useful when time is not given or not needed.

Summary of All Four Kinematic Equations

Equation Formula Missing Variable Best Used When
1 v = v₀ + a t Δx Time is known
2 Δx = (v + v₀)/2 × t a Velocities known
3 Δx = v₀ t + ½ a t² v Time and acceleration known
4 v² = v₀² + 2 a Δx t Time is unknown

Frequently Asked Questions

Why do we need to derive the kinematic equations?

Derivations help you understand the logic behind the formulas instead of just memorizing them.

Are these derivations only for constant acceleration?

Yes. All four equations assume acceleration is constant.

Which equation is derived last?

Equation 4 is usually derived last because it eliminates time.

Continue Mastering Kinematics on physicalfundamentals.info:

Main Kinematic Equations Guide 1D Kinematics Equations 2D Kinematics & Projectile Motion What Is Kinematics? ← Back to Physics Fundamentals

Now that you understand the derivations, you will use the kinematic equations with much more confidence. The algebra behind them is straightforward once you know the starting definitions.

Last updated: April 2026 | Written for students by physics educators at physicalfundamentals.info