The Second Equation of Motion (s = ut + ½at²)
Simple Explanation
The second equation of motion, s = ut + ½at², lets you calculate the displacement (s) of an object given its initial velocity (u), acceleration (a), and time (t).
Why Do We Need It?
This equation lets you find how far something travels during acceleration, without needing to know its final velocity first.
Formula
Equation of Motion: s = ut + ½at²
s = ut + ½at²
Displacement from initial velocity, acceleration, and time.
- s
- — displacement (m)
- u
- — initial velocity (m/s)
- a
- — acceleration (m/s²)
- t
- — time (s)
When to use it: Whenever you know initial velocity, acceleration and time, and need the distance travelled.
Worked Example
Find the distance a ball travels
A ball starts at 2 m/s and accelerates at 4 m/s² for 3 s. Find the distance it travels.
Why Does This Work?
The first term (ut) accounts for the distance the object would cover at its constant starting speed; the second term (½at²) adds the extra distance gained purely from accelerating — together they give the true total displacement.
Real-Life Example
Calculating runway length for takeoff
Aircraft engineers calculate how much runway a plane needs to reach takeoff speed.
Given the plane's acceleration and the time it takes to reach takeoff speed, this equation gives exactly how much distance (runway) is needed.
Practice
An object starts at 0 m/s and accelerates at 2 m/s² for 5 s. What distance does it travel?
MediumCommon mistake
Forgetting to square the time before multiplying by ½a — the t² applies only to time, not to the whole ½at term.
Quick Review
- s = ut + ½at²
- s = displacement, u = initial velocity, a = acceleration, t = time.
- Useful when you know time but not final velocity.