3bKinematic Equations
Learn the four equations of motion, develop a strategy for choosing the right one, and solve constant-acceleration problems confidently.
The kinematic equations are the core toolkit of mechanics. With just five variables and four equations, you can solve virtually any constant-acceleration problem — from car crashes to rocket launches.
Lesson Overview
The four kinematic equations relate displacement, initial velocity, final velocity, acceleration, and time for objects undergoing constant acceleration. In this lesson you will learn each equation, identify which variables each contains, and develop a strategy for choosing the right equation to solve any kinematics problem.
Key Concepts
Equation 1
v = v₀ + at — links final velocity, initial velocity, acceleration, and time (no displacement)
Equation 2
x = v₀t + ½at² — links displacement, initial velocity, acceleration, and time (no final velocity)
Equation 3
v² = v₀² + 2ax — links final velocity, initial velocity, acceleration, and displacement (no time)
Equation 4
x = ½(v + v₀)t — links displacement, average velocity, and time (no acceleration)
Choosing an Equation
List known and unknown variables; pick the equation that contains exactly those variables
Constant Acceleration Only
All four equations assume acceleration is constant throughout the motion
A car starts from rest and accelerates at 3 m/s² for 8 s. Find its final velocity.
A ball rolls from rest with a = 2 m/s² for 5 s. How far does it travel?
A car traveling at 20 m/s brakes with a = −4 m/s². How far does it travel before stopping?
An object accelerates from 4 m/s to 16 m/s over a distance of 30 m. Find the acceleration.
A train accelerates from 10 m/s to 30 m/s in 8 s. Find the displacement using Equation 4.
A motorcycle starts from rest and reaches 18 m/s in 6 s. Find its acceleration.
Hint: You know v₀, v, and t but not x. Which equation uses only those four variables?
A skier starts from rest and accelerates at 1.5 m/s² down a slope for 10 s. How far does she travel?
Hint: You know v₀, a, and t. Which equation gives displacement without needing final velocity?
A ball is thrown at 12 m/s and decelerates at 3 m/s². How far does it travel before stopping?
Hint: You know v₀, v (= 0), and a. Which equation connects these to displacement without time?
A car accelerates from 5 m/s to 25 m/s over 100 m. How long does this take?
Hint: You know v₀, v, and x. Find a first using v² = v₀² + 2ax, then use v = v₀ + at to find t.
Why can't you use the kinematic equations for a car that is speeding up and then braking in the same problem?
Hint: Think about what assumption all four equations require about acceleration.
Key Vocabulary
Kinematic Equations
A set of four equations relating displacement, initial velocity, final velocity, acceleration, and time for constant acceleration.
Example: Using v² = v₀² + 2ax to find stopping distance without knowing time.
Displacement (x)
The change in position of an object; a vector quantity with magnitude and direction.
Example: A car that moves 50 m forward has a displacement of +50 m.
Initial Velocity (v₀)
The velocity of an object at the start of the time interval being analyzed.
Example: A ball thrown at 10 m/s has v₀ = 10 m/s.
Constant Acceleration
Acceleration that does not change in magnitude or direction over the time interval.
Example: A freely falling object near Earth's surface has constant acceleration of 9.8 m/s² downward.
Interactive Practice — 5 Questions
Which kinematic equation would you use if you know v₀, a, and t, and want to find displacement?
A car starts from rest with a = 5 m/s² for 4 s. What is its final velocity?
Which equation has NO time variable?
The kinematic equations are valid only when acceleration is:
An object with v₀ = 6 m/s, a = 2 m/s², travels for 3 s. What is its displacement?
Independent Practice
A rocket starts from rest and accelerates at 12 m/s² for 5 s. Find (a) its final velocity and (b) its displacement.
A car traveling at 25 m/s brakes with a = −5 m/s². How far does it travel before stopping? How long does it take?
An object moves from 3 m/s to 15 m/s over a displacement of 36 m. Find the acceleration.
List all four kinematic equations and state which variable is missing from each one.
★ A ball is launched horizontally at 8 m/s from a 20 m cliff. Using kinematics for the vertical direction (a = 9.8 m/s², v₀ = 0), find the time to hit the ground and the horizontal distance traveled.
ChallengeCommon Mistakes
Using kinematic equations when acceleration is not constant (e.g., a car with varying engine force).
Kinematic equations only apply to constant acceleration. For varying acceleration, calculus or numerical methods are needed.
Forgetting to set v₀ = 0 when an object starts from rest.
Always explicitly write v₀ = 0 for objects starting from rest — it simplifies the equation and prevents errors.
Math Tips
Make a "VUVAT" table: list v, u (v₀), a, t, and x. Mark what you know (✓) and what you want (?). The equation that contains only your knowns and your unknown is the one to use.