Kinematics Equations Cheat Sheet With Worked Problems

Overview

Kinematics is the branch of mechanics that describes motion without focusing on the forces that cause it. In most school-level physics, kinematics problems ask you to connect quantities such as displacement, velocity, acceleration, and time. The challenge is usually not algebra alone. The real difficulty is identifying which equation matches the information you are given.

This reference focuses on the standard constant-acceleration equations, sometimes called the SUVAT equations. They are most useful when acceleration stays constant during the motion. That includes common textbook cases such as a car speeding up evenly, an object in free fall near Earth when air resistance is ignored, or a runner starting from rest and accelerating steadily.

Before using the formulas, keep these symbols straight:

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

The core kinematics equations cheat sheet is:

• v = u + at
• s = ut + 1/2 at²
• v² = u² + 2as
• s = ((u + v) / 2)t

Some courses also use:

• s = vt - 1/2 at²

That last form comes from rearranging the others, so it is less essential to memorize if you already know the main four.

A good first question for any motion problem is: Is the acceleration constant? If the answer is no, these formulas may not apply directly. If the answer is yes, the cheat sheet becomes very useful.

Core concepts

This section explains the ideas behind the equations so they feel less like disconnected formulas.

1. Displacement is not the same as distance

Displacement measures change in position in a chosen direction. Distance is the total path length traveled. In one-dimensional kinematics, equations usually use displacement, not distance. That matters when direction changes. If a student walks 3 m east and then 3 m west, the total distance is 6 m but the displacement is 0 m.

2. Velocity is directional

Velocity includes speed and direction. A velocity of +5 m/s means motion in the positive direction you defined. A velocity of -5 m/s means motion in the opposite direction. Sign mistakes are one of the most common sources of lost marks in motion equations practice.

3. Acceleration tells you how velocity changes

Acceleration is the rate of change of velocity. Positive acceleration does not always mean an object is speeding up. It only means the acceleration points in the positive direction. If velocity and acceleration have opposite signs, the object is slowing down.

4. Choose a sign convention and keep it consistent

For vertical motion, many students choose upward as positive. In that case, acceleration due to gravity is negative, so a = -9.8 m/s² or sometimes -9.81 m/s². Other students choose downward as positive. That also works, but then gravity becomes positive. The key is consistency.

5. Each equation leaves out one variable

This is the easiest way to decide which formula to use:

• v = u + at does not include s
• s = ut + 1/2 at² does not include v
• v² = u² + 2as does not include t
• s = ((u + v) / 2)t does not include a

If a question gives you three variables and asks for a fourth, look for the equation that does not include anything extra.

6. Units matter

Use standard SI units unless your course says otherwise:

• displacement in meters (m)
• velocity in meters per second (m/s)
• acceleration in meters per second squared (m/s²)
• time in seconds (s)

Convert before solving. For example, 72 km/h should become 20 m/s before substitution.

7. The equations come from simple relationships

Knowing the ideas behind the equations can help you remember them:

• v = u + at says final velocity equals initial velocity plus change in velocity
• s = ((u + v) / 2)t says displacement equals average velocity multiplied by time
• s = ut + 1/2 at² combines starting motion with extra displacement gained from accelerating

That conceptual view often helps more than memorizing symbol patterns.

Worked problem 1: finding final velocity

Question: A bicycle starts at 4 m/s and accelerates at 1.5 m/s² for 6 s. What is its final velocity?

Given: u = 4 m/s, a = 1.5 m/s², t = 6 s

Find: v

Choose equation: v = u + at

Substitute: v = 4 + (1.5 × 6)

Calculate: v = 4 + 9 = 13 m/s

Answer: The final velocity is 13 m/s.

Worked problem 2: finding displacement from rest

Question: A car starts from rest and accelerates uniformly at 2 m/s² for 5 s. How far does it travel?

Given: u = 0 m/s, a = 2 m/s², t = 5 s

Find: s

Choose equation: s = ut + 1/2 at²

Substitute: s = (0 × 5) + 1/2 × 2 × 5²

Calculate: s = 0 + 1 × 25 = 25 m

Answer: The car travels 25 m.

Worked problem 3: solving without time

Question: A train increases its speed from 10 m/s to 18 m/s over a displacement of 56 m. What is its acceleration?

Given: u = 10 m/s, v = 18 m/s, s = 56 m

Find: a

Choose equation: v² = u² + 2as

Substitute: 18² = 10² + 2a(56)

Calculate: 324 = 100 + 112a

224 = 112a

a = 2 m/s²

Answer: The acceleration is 2 m/s².

Worked problem 4: vertical motion under gravity

Question: A ball is thrown upward with an initial velocity of 20 m/s. How long does it take to reach its highest point?

Idea: At the highest point, final velocity is 0 m/s.

Given: u = 20 m/s, v = 0 m/s, a = -9.8 m/s²

Find: t

Choose equation: v = u + at

Substitute: 0 = 20 - 9.8t

Calculate: 9.8t = 20

t ≈ 2.04 s

Answer: It takes about 2.0 s to reach the highest point.

If you are also studying why motion changes in the first place, it helps to connect kinematics with force ideas in Newton’s Laws of Motion Study Guide With Real-World Examples and Practice.

Related terms

These terms often appear alongside kinematics formulas explained in class notes and exam questions.

Scalar vs vector

A scalar has magnitude only, such as speed or distance. A vector has magnitude and direction, such as velocity, displacement, and acceleration. Confusing speed with velocity is a common reason students choose the wrong sign or the wrong formula.

Average speed vs average velocity

Average speed = total distance ÷ total time. Average velocity = displacement ÷ time. They are only equal when motion stays in one direction and the distinction between distance and displacement does not matter.

Uniform velocity

Uniform velocity means constant velocity: both speed and direction stay the same. If velocity is truly constant, acceleration is zero. Then the displacement formula simplifies to s = vt.

Free fall

Free fall describes motion when gravity is the only significant force acting. Near Earth’s surface and ignoring air resistance, acceleration is treated as constant. That is why the standard kinematics formulas work well for many vertical motion problems.

Graph links

Kinematics is easier when you connect equations to graphs:

• The slope of a displacement-time graph gives velocity
• The slope of a velocity-time graph gives acceleration
• The area under a velocity-time graph gives displacement

Even when a question looks algebraic, a quick sketch can clarify what is happening.

Rest and turnaround points

An object can have zero velocity at an instant without being permanently at rest. For example, a ball thrown upward has zero velocity at the top of its path, but gravity is still acting, so it immediately begins moving downward. That distinction matters in multi-step questions.

Constant acceleration limitations

The cheat sheet is powerful, but it has limits. If acceleration changes continuously, as in complicated air resistance situations, you may need graphs, calculus-based methods, or piecewise analysis. In most school-level problems, however, the motion is set up to make acceleration constant.

Practical use cases

Here is how to use this reference effectively for homework help, revision, and test prep.

Use case 1: selecting the right equation fast

Write down the variables you know and the variable you need. Then choose the equation that contains those quantities and omits the one you do not know. This simple screening method saves time and reduces equation-hopping.

Example: If you know u, a, and t and need s, use s = ut + 1/2 at². Do not start with v² = u² + 2as because it introduces another unknown, v.

Use case 2: solving multi-step problems

Some problems need more than one equation. For example, if you are asked how high a ball rises, you may first identify that at maximum height v = 0, then use v² = u² + 2as to solve for displacement. If the question also asks total time in the air, you might then use a second equation for another part of the motion.

Example: A ball is thrown upward at 14 m/s. Find maximum height.

Given: u = 14 m/s, v = 0, a = -9.8 m/s²

Use: v² = u² + 2as

0 = 14² + 2(-9.8)s

0 = 196 - 19.6s

19.6s = 196

s = 10 m

The maximum height above the release point is 10 m.

Use case 3: checking whether an answer makes sense

Physics answers should be plausible. If a car accelerating gently for 3 seconds gives you a final speed of 300 m/s, something has gone wrong. After solving, check:

• Are the units correct?
• Is the sign reasonable?
• Is the size of the answer realistic?
• Does the answer fit the story in the question?

This habit catches many arithmetic and sign errors before you hand in work.

Use case 4: working with free-fall questions

For vertical motion, define positive direction first. Then assign gravity the correct sign. Many students lose marks not because they do not know the formulas, but because they switch sign conventions halfway through.

Example: A stone is dropped from rest and falls for 3 s. How far does it fall?

Take downward as positive. Then u = 0, a = 9.8 m/s², t = 3 s.

s = ut + 1/2 at²

s = 0 + 1/2 × 9.8 × 9

s = 44.1 m

So it falls 44.1 m.

If you chose upward as positive instead, you would use a negative acceleration and a negative displacement. Both approaches can work if you stay consistent.

Use case 5: exam revision and formula memory

To remember the equations, do not only recite them. Practice matching each formula to a problem type:

• v = u + at: velocity after a time
• s = ut + 1/2 at²: displacement when time is known
• v² = u² + 2as: displacement or acceleration when time is missing
• s = ((u + v) / 2)t: displacement from average velocity

That method turns memorization into recognition.

Common mistakes to avoid

• Using distance when the equation requires displacement
• Forgetting that velocities can be negative
• Mixing kilometers per hour with meters per second
• Using constant-acceleration equations when acceleration is not constant
• Dropping the square on t² or v²
• Confusing initial velocity u with final velocity v

A quick problem-solving checklist

1. Sketch the motion if possible
2. List known variables with units
3. Choose a positive direction
4. Select the equation that fits the knowns and unknown
5. Substitute carefully with signs
6. Solve algebraically
7. Check units and reasonableness

This is the kind of repeatable routine that makes a physics study guide genuinely useful rather than just decorative.

When to revisit

Revisit this cheat sheet whenever the inputs in a motion problem change, especially if you are not sure which equation to pick. It is most useful in these moments:

• Before a quiz on motion, SUVAT, or free fall
• When homework gives you a new combination of known and unknown variables
• When you keep getting the algebra right but the setup wrong
• When your class moves from horizontal motion to vertical motion
• When you start graph-based kinematics and want to connect formulas to meaning
• When you need quick revision for GCSE science revision, introductory college physics, or AP-style mechanics review

A practical way to use this page is to turn it into a personal reference. Add one solved example of your own under each equation. Include one horizontal problem and one vertical problem. Then, when you revisit, you are not starting from zero; you are returning to examples you already understand.

You can also revisit after marking a test. Sort your mistakes into categories: equation choice, unit conversion, signs, algebra, or interpretation. Then use the matching part of this guide to patch that specific weakness. That is more effective than rereading every chapter from the start.

Finally, if your next physics topic is dynamics, circular motion, or energy, keep this page nearby. Many later topics still depend on clear motion descriptions. Kinematics is often the language those later problems are written in. The better you read that language, the easier the rest of physics becomes.

Action step: Take three problems from your notes and label each by the equation it needs before you solve it. If you can choose the formula confidently, you are already making progress.