Kinematic Equations — The 4 SUVAT Formulas Explained

The kinematic equations are a set of formulas for motion with constant acceleration. If acceleration changes during the motion, these formulas do not apply in their standard form.

In many courses, they are called the SUVAT equations because they connect five variables:

• $s$: displacement
• $u$: initial velocity
• $v$: final velocity
• $a$: acceleration
• $t$: time

If you know three of these and the acceleration is constant, one of the equations will often give you the fourth directly.

The 4 Standard SUVAT Formulas

For one-dimensional motion with constant acceleration:

$$v = u + at$$ $$s = ut + \frac{1}{2}at^2$$ $$v^2 = u^2 + 2as$$ $$s = \frac{u + v}{2}t$$

These formulas are closely related, so you do not need to memorize them as separate facts without context. Each one is just a different way to connect the same motion variables.

What Each Formula Is Good For

Use $v = u + at$ when you want to connect speed change to time.

Use $s = ut + \frac{1}{2}at^2$ when you need displacement over a known time interval.

Use $v^2 = u^2 + 2as$ when time is not given and you want to avoid solving for it.

Use $s = \frac{u + v}{2}t$ when you know the initial and final velocities and want displacement from the average velocity. This works here because, under constant acceleration, velocity changes linearly with time.

The Condition That Matters Most

The most common mistake is using these equations when acceleration is not constant.

For example, they work well for idealized free fall near Earth's surface if you ignore air resistance, because the acceleration can be treated as constant. They do not directly describe motion where acceleration depends strongly on time, speed, or position unless you break the motion into simpler pieces or use calculus.

One Worked Example

A car starts from rest and accelerates in a straight line at $2\ \mathrm{m/s^2}$ for $5\ \mathrm{s}$. Find its final velocity and displacement.

Here, the known values are

$$u = 0,\quad a = 2\ \mathrm{m/s^2},\quad t = 5\ \mathrm{s}$$

First find the final velocity:

$$v = u + at = 0 + (2)(5) = 10\ \mathrm{m/s}$$

Now find the displacement:

$$s = ut + \frac{1}{2}at^2$$ $$s = 0 + \frac{1}{2}(2)(5^2) = 25\ \mathrm{m}$$

So after $5$ seconds, the car is moving at $10\ \mathrm{m/s}$ and has traveled $25\ \mathrm{m}$.

This example is simple, but it shows the practical pattern: identify what is known, pick the equation that matches, and solve only for the variable you need.

Common Mistakes

Mixing up velocity and acceleration

Velocity tells you how fast position changes. Acceleration tells you how fast velocity changes. If those roles are mixed, the equation choice usually goes wrong immediately.

Ignoring the sign convention

Direction matters. If upward is positive, then downward acceleration in free fall must be negative. A wrong sign can produce an answer that looks numerically correct in size but is physically wrong.

Using the wrong formula for the known data

If time is not given, $v^2 = u^2 + 2as$ is often the cleanest choice. Solving with a formula that introduces an extra unknown usually creates unnecessary algebra.

Treating distance and displacement as the same thing

In these equations, $s$ is displacement, not total path length. That distinction matters whenever direction changes.

Where Kinematic Equations Are Used

These formulas show up in basic mechanics, vehicle motion, projectile motion split into horizontal and vertical components, and free-fall problems. They are especially useful at the stage where you want to describe motion directly, before bringing in Newton's laws or energy methods.

They are also a good test of physical understanding: if you can tell which variable is missing and which equation avoids it, the problem usually becomes much easier.

A Practical Next Step

Try your own version of the example by changing just one value, such as making the acceleration $3\ \mathrm{m/s^2}$ or the time $4\ \mathrm{s}$, and predict what should happen before you calculate it. If you want to explore another constant-acceleration case with your own numbers, solve a similar problem with GPAI Solver.