Kinematic Equations — The 4 SUVAT Formulas Explained

The kinematic equations are a set of formulas for motion with constant acceleration, and the procedure for using them is almost always the same: check the condition, list what you know, pick the equation that avoids the unknown you do not have, then solve. 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 equation will often give you the fourth directly.

When This Method Applies

Use the SUVAT approach only when the acceleration is constant over the interval you are analyzing, and the motion is treated in one dimension (or one component of it). 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.

The Procedure, Step By Step

Step 1: Check the condition and fix a sign convention

Confirm the acceleration is constant and decide which direction is positive. If upward is positive, downward acceleration in free fall must be negative.

Step 2: List the known variables

Identify which of $s$, $u$, $v$, $a$, and $t$ are given and which one you need to find.

Step 3: Pick the formula that matches your knowns

For one-dimensional motion with constant acceleration, the four standard relations are

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

Choose by what you have and what you want to avoid:

• $v = u + at$ when you want to connect speed change to time.
• $s = ut + \frac{1}{2}at^2$ when you need displacement over a known time interval.
• $v^2 = u^2 + 2as$ when time is not given and you want to avoid solving for it.
• $s = \frac{u + v}{2}t$ when you know the initial and final velocities and want displacement from the average velocity. This works because, under constant acceleration, velocity changes linearly with time.

Step 4: Substitute with units and signs, then check

Confirm the answer matches the physical motion you expect.

Worked Example: The Whole Procedure At Once

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.

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:

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

So after $5$ seconds the car is moving at $10\ \mathrm{m/s}$ and has traveled $25\ \mathrm{m}$. The pattern is the whole point: identify what is known, pick the equation that matches, and solve only for the variable you need.

Where Each Step Tends To Break, And How To Check

Mixing up velocity and acceleration

Velocity tells you how fast position changes; acceleration tells you how fast velocity changes. If those roles get mixed at Step 2, the equation choice usually goes wrong immediately.

Ignoring the sign convention

A wrong sign can produce an answer that looks numerically right in size but is physically wrong. Re-check the direction you chose as positive in Step 1.

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 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. Self-check: did the object reverse direction during the interval?

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.

Predict, Then Verify

Take the example and change one value, such as making the acceleration $3\ \mathrm{m/s^2}$ or the time $4\ \mathrm{s}$. Predict whether the displacement should grow or shrink before you calculate, then run the procedure and check whether your prediction held.