The Big Five Kinematics Equations- Essential Physics Formulas
What the Big Five Kinematics Equations Actually Are
These five formulas describe motion with constant acceleration. That's it. They're not mysterious, they're not optional, and if you're taking physics, you'll use them until they're burned into your brain.
The equations relate four variables: initial velocity (v₀), final velocity (v), acceleration (a), time (t), and displacement (s). Master these five, and you can solve virtually any constant acceleration problem thrown at you.
The Big Five
Equation 1: v = v₀ + at
Velocity equals initial velocity plus acceleration times time. This is the most straightforward one.
Use when: You know v₀, a, and t — and need to find final velocity.
Equation 2: s = v₀t + ½at²
Displacement equals initial velocity times time plus half of acceleration times time squared.
Use when: You know v₀, a, and t — and need to find displacement. This equation ignores final velocity entirely.
Equation 3: v² = v₀² + 2as
Final velocity squared equals initial velocity squared plus two times acceleration times displacement.
Use when: Time isn't given and you don't need it. This equation skips t entirely, which is useful when the problem doesn't mention time.
Equation 4: s = ½(v + v₀)t
Displacement equals half the sum of velocities, times time. This is the average velocity formula disguised as a kinematics equation.
Use when: You know both velocities and time, but not acceleration. Average velocity times time gives you displacement.
Equation 5: s = vt - ½at²
Displacement equals final velocity times time minus half acceleration times time squared.
Use when: You know v, a, and t — but not initial velocity. It's algebraically equivalent to equation 2, just rearranged.
The Comparison Table
| Equation | Formula | Known Variables | Solves For |
|---|---|---|---|
| 1 | v = v₀ + at | v₀, a, t | v |
| 2 | s = v₀t + ½at² | v₀, a, t | s |
| 3 | v² = v₀² + 2as | v₀, a, s | v |
| 4 | s = ½(v + v₀)t | v, v₀, t | s |
| 5 | s = vt - ½at² | v, a, t | s |
How to Actually Use These Equations
Most students memorize the formulas and then freeze up when they see a problem. Here's the real process:
- List what you know. Write down v₀, v, a, t, and s from the problem. Put a question mark next to what you're solving for.
- Find the equation. Pick the equation that contains your known variables and your unknown. If time isn't given, use equation 3. If initial velocity isn't given, use equation 5.
- Plug in the numbers. Watch your units. Meters and seconds, or feet and seconds — stay consistent.
- Solve. Algebra is your friend here. Isolate the unknown.
Practical Example
A car accelerates from rest (v₀ = 0) at 4 m/s² for 6 seconds. How far does it travel?
Known: v₀ = 0, a = 4 m/s², t = 6 s. Unknown: s.
Equation 2 works here: s = v₀t + ½at²
s = (0)(6) + ½(4)(6)²
s = 0 + ½(4)(36)
s = 72 meters
That's it. That's the whole process.
Common Mistakes That Cost Points
- Using the wrong sign for acceleration. Deceleration is negative acceleration. If something slows down, a is negative in your equation.
- Confusing velocity and displacement. These are different variables. Don't swap them.
- Forgetting to square the time. In equation 2, it's t². Students lose marks on this constantly.
- Using these equations when acceleration isn't constant. They don't work for changing acceleration. You'd need calculus for that.
Which Equation Do I Pick?
If time isn't mentioned: Equation 3 (v² = v₀² + 2as).
If initial velocity isn't given: Equation 5 (s = vt - ½at²).
If final velocity isn't given: Equation 2 (s = v₀t + ½at²).
If you only have velocities and time: Equation 4 (s = ½(v + v₀)t).
If you need velocity and have time: Equation 1 (v = v₀ + at).
Pick the equation that matches what you have. That's the entire strategy.