The Third Equation of Motion (v² = u² + 2as)
Simple Explanation
The third equation of motion, v² = u² + 2as, relates final velocity (v), initial velocity (u), acceleration (a), and displacement (s) — without needing to know the time taken.
Why Do We Need It?
This is the equation to reach for whenever a problem gives you distance instead of time — it lets you skip straight to velocity or acceleration without solving for time first.
Formula
Equation of Motion: v² = u² + 2as
v² = u² + 2as
Final velocity from initial velocity, acceleration, and displacement — no time needed.
- v
- — final velocity (m/s)
- u
- — initial velocity (m/s)
- a
- — acceleration (m/s²)
- s
- — displacement (m)
When to use it: Whenever time is unknown but displacement is, and you need velocity or acceleration.
Worked Example
Find a car's final velocity from distance
A car starts at 4 m/s and accelerates at 2 m/s² over a distance of 24 m. Find its final velocity.
Why Does This Work?
This equation can be derived by combining the first two equations of motion to eliminate time altogether — which is exactly why it's the one to use when time isn't given.
Real-Life Example
Calculating a car's stopping distance
Road safety calculations find a car's speed just before a collision, using its deceleration and the length of its skid marks (a measurable displacement).
Since the skid distance is known but not the time, this equation is exactly the tool used to work backward to the speed.
Practice
An object starts at 0 m/s and accelerates at 5 m/s² over a distance of 10 m. Find its final velocity.
HardCommon mistake
Forgetting the final square-root step — the equation gives v², not v, so you must take the square root to find the actual velocity.
Quick Review
- v² = u² + 2as
- Use this equation when time is unknown or not needed.
- Remember to take the square root of the result to find v.