What does 'seconds squared' mean in acceleration units (m/s²)?

If you’ve ever wondered what “square seconds,” s², mean in the unit of acceleration (m/s²), you’re not alone. Square metres (m²) are easy to picture, since we use them in everyday situations, such as measuring the area of a room or a rug. And because space is three-dimensional, we can even cube the metre (m³) without much confusion. But squaring a one-dimensional quantity like time can definitely raise eyebrows. Why would we want to square time, and what does it mean? For instance, if a car accelerates at a = 2 m/s², how much would its speed increase in 2 seconds? Do we square the seconds to find the answer?

No, we don't literally square time to calculate acceleration. The s² can look strange at first, but it has a simple explanation. Measurement systems are designed to be compact, using as few base units as possible. With just metres and seconds, we can describe many important physical quantities, such as length, time, speed, and acceleration. As a result, units sometimes “stack,” as they do for acceleration, giving the slightly odd-looking m/s², which you can read as “metres per second per second.” Let’s take it step by step and see what this means in practical terms.

Speed: how length changes over time

Speed (v) tells us how fast the position changes with time. If a car moving at a constant speed covers a distance Δd = 30 meters in a time interval Δt = 3 seconds, we can calculate its speed as: i

v = Δd ÷ Δt = 30 m ÷ 3 s = 10 m/s.

Acceleration: how speed changes over time

Acceleration (a) tells us how fast the speed changes with time. Our car, which was previously moving at a constant speed of v = 10 m/s, begins to accelerate and reaches 20 m/s in 5 seconds. We can determine its acceleration as:

a = Δv ÷ Δt = (20−10) m/s ÷ 5 s = 10 m/s ÷ 5 s =2 m/s/s = 2 m/s²

As you can see, we didn’t use any square numbers; we divided speed by time. The squared term, s², appears because speed already contains a time unit. In other words, the sxs=s² comes from two different roles of time: one second comes from measuring speed (m/s), and the other comes from measuring how that speed changes per second.

Calculating a change in speed due to acceleration

Let’s summarize what we’ve learned. Initially, our car was traveling at a constant speed of 10 m/s. It then began to accelerate at a = 2 m/s². This means the car increases its speed by 2 m/s every second:

• After 1 second, the speed: v = 10 m/s + 2 m/s = 12 m/s

• After 2 seconds, the speed: v = 12 m/s + 2 m/s = 14 m/s

• After 3 seconds, the speed v: = 14 m/s + 2 m/s = 16 m/s

Each second, the speed grows by 2 m/s. This repeated change per second is exactly what gives rise to the 'squared' time unit in acceleration.