Kinematic Equations
Solve one-dimensional constant-acceleration motion, free fall, projectile motion, displacement, velocity, acceleration, time, and motion-table problems. Enter the known values, choose the unknown, and get the formula, result, units, graph-style diagram, and full explanation.
Kinematic Equations Calculator
Select a mode below. The main solver works for constant acceleration in one dimension. The projectile calculator splits the launch velocity into horizontal and vertical components. The free-fall calculator uses vertical acceleration due to gravity. All results are estimates for ideal physics problems unless you add real-world corrections separately.
Known Values
Distance and Time
1D Motion Result
Free Fall Inputs
Gravity and Direction
Free Fall Result
Projectile Inputs
Gravity
Projectile Result
| Point | Time \(t\) | Horizontal position \(x\) | Vertical position \(y\) |
|---|---|---|---|
| Trajectory table will appear here. | |||
Motion Table Inputs
Table Formula
This mode helps students see how velocity and displacement change at equal time intervals when acceleration is constant.
Generated Motion Table
| Time \(t\) | Velocity \(v\) | Displacement \(s\) | Interpretation |
|---|---|---|---|
| Click generate table. | |||
What Are Kinematic Equations?
Kinematic equations are mathematical relationships used to describe motion without directly studying the forces that caused the motion. In school physics, the phrase usually refers to the constant-acceleration equations that connect five main variables: displacement, initial velocity, final velocity, acceleration, and time. These variables are often written as \(s, u, v, a,\text{ and }t\). Some textbooks use \(x, x_0, v_0, v,\text{ and }a\). The symbols can change, but the physical meaning remains the same.
Kinematics is the first major language of motion. Before a student studies force, energy, momentum, rotation, fields, or waves, they need to understand how position changes with time. Kinematic equations make this possible because they allow us to calculate missing motion quantities from known quantities. For example, if a car starts from rest, accelerates at a constant rate, and moves for a known time, we can calculate its final velocity and displacement. If a ball is thrown upward and gravity is the only acceleration, we can calculate how long it stays in the air and how high it rises.
The key condition is constant acceleration. These equations are exact only when acceleration does not change during the interval. This is why they work well for many introductory problems: a vehicle accelerating uniformly, a dropped object near Earth when air resistance is ignored, a ball thrown vertically, or a projectile moving under ideal gravity. They are not automatically valid for motion with changing acceleration, such as a rocket burning fuel, a car with variable throttle, a parachutist with strong air drag, or an object on a spring. In those cases, calculus, numerical simulation, or graphs may be needed.
The Five Main Variables
Every basic kinematic problem is built from five quantities. Displacement \(s\) is the change in position. It is not always the same as distance. Distance is the total path length traveled, while displacement includes direction. If you walk 10 metres east and then 10 metres west, your distance is 20 metres but your displacement is 0 metres. In one-dimensional kinematics, displacement can be positive, negative, or zero.
Initial velocity \(u\) is the velocity at the start of the time interval. Final velocity \(v\) is the velocity at the end of the interval. Acceleration \(a\) is the rate of change of velocity with time. Time \(t\) is the duration of the motion. These variables must use consistent units. The standard SI set is metres, seconds, metres per second, and metres per second squared.
| Symbol | Name | SI unit | Meaning |
|---|---|---|---|
| \(s\) | Displacement | m | Change in position from start to finish. |
| \(u\) | Initial velocity | m/s | Velocity at the beginning of the interval. |
| \(v\) | Final velocity | m/s | Velocity at the end of the interval. |
| \(a\) | Acceleration | m/s² | Rate at which velocity changes. |
| \(t\) | Time | s | Duration of the motion. |
The Core Kinematic Equations
The four standard equations for constant acceleration are:
Each equation contains four of the five main variables. This is useful because many problems give three known values and ask for a fourth. If time is not given, use an equation without time. If final velocity is not given, use an equation without final velocity. Choosing the correct equation is often the most important step in solving a kinematics problem.
How to Choose the Right Equation
The simplest method is to list the knowns and the unknown. Suppose a car starts from rest with \(u=0\), accelerates at \(a=3\,m/s^2\), and moves for \(t=10\,s\). If the question asks for displacement, the equation \(s=ut+\frac{1}{2}at^2\) is the best choice because it uses \(u\), \(a\), and \(t\), and solves for \(s\). If the question asks for final velocity, use \(v=u+at\). If the problem gives displacement and acceleration but not time, then \(v^2=u^2+2as\) may be better.
A common student mistake is trying to force every problem into one equation. The equations are a toolkit. The right tool depends on the missing variable. Another common mistake is ignoring signs. If upward is positive, gravity is negative. If right is positive, motion to the left is negative. Signs are not decoration; they carry direction.
Deriving the First Equation
Acceleration is defined as the rate of change of velocity:
Rearranging gives:
This equation says that final velocity equals initial velocity plus the velocity gained from acceleration over time. If acceleration is positive, velocity increases in the positive direction. If acceleration is negative, velocity decreases or changes toward the negative direction. If acceleration is zero, the equation becomes \(v=u\), meaning velocity stays constant.
Deriving Displacement Under Constant Acceleration
When acceleration is constant, average velocity can be found by taking the mean of initial and final velocities:
Substituting \(v=u+at\) into this equation gives:
This is one of the most useful kinematic equations because it directly connects displacement with initial velocity, acceleration, and time. The term \(ut\) represents displacement that would happen if the object continued at its initial velocity. The term \(\frac{1}{2}at^2\) represents the extra displacement caused by acceleration.
The Equation Without Time
Sometimes time is not known and not required. In that case, the equation
is often the best choice. It is especially useful for stopping distance, runway acceleration, vertical motion, and free-fall speed. For example, if an object is dropped from height \(h\) with \(u=0\), then \(v^2=2gh\), so \(v=\sqrt{2gh}\). This gives the ideal impact speed without first calculating time.
Free Fall Motion
Free fall is vertical motion under gravity alone. In ideal free fall, air resistance is ignored and the acceleration is constant near Earth’s surface. Many physics courses use \(g=9.8\,m/s^2\) as the magnitude of gravitational acceleration. The sign depends on your coordinate system. If upward is positive, then \(a=-g\). If downward is positive, then \(a=+g\).
If an object is dropped from rest, then \(u=0\). The equations simplify:
These formulas are powerful, but they are idealized. A feather, paper sheet, or parachutist does not follow simple free-fall equations for long because air resistance becomes important. A dense ball dropped from a modest height may be close enough for school-level calculations.
Projectile Motion
Projectile motion combines horizontal and vertical motion. The horizontal component usually has zero acceleration if air resistance is ignored. The vertical component has constant acceleration due to gravity. The first step is to split the initial velocity into components:
Horizontal position is:
Vertical position is:
If the projectile lands at the same height from which it was launched, the common range and maximum-height formulas are:
If the projectile launches from a height above the landing level, the flight time comes from solving a quadratic equation. This calculator handles launch height by solving:
Velocity-Time Graphs
Kinematic equations also connect closely to graphs. On a velocity-time graph, acceleration is the slope and displacement is the area under the graph. If acceleration is constant, the velocity-time graph is a straight line. The area under the line forms a rectangle plus a triangle. That is why the displacement formula contains \(ut\) and \(\frac{1}{2}at^2\).
On a displacement-time graph, the slope gives velocity. If velocity is constant, the graph is a straight line. If acceleration is constant but not zero, the displacement-time graph is curved because displacement depends on \(t^2\). On an acceleration-time graph, the area gives change in velocity.
Common Kinematics Problem Types
1. Starting From Rest
“Starting from rest” means \(u=0\). This phrase appears often in car, elevator, train, ball, and runway problems. The equations become simpler:
2. Stopping Motion
Stopping means final velocity is \(v=0\). If a car brakes from speed \(u\) with constant deceleration, stopping distance can be found using:
Because braking acceleration is negative when forward is positive, the final distance becomes positive.
3. Vertical Throw
When an object is thrown upward, gravity slows it down until its velocity becomes zero at the top. At maximum height:
4. Average Velocity
Under constant acceleration, average velocity is:
This formula only works directly when acceleration is constant. For non-uniform acceleration, average velocity must be calculated from total displacement divided by total time.
Unit Conversion in Kinematics
Many wrong answers in kinematics come from mixed units. If velocity is in kilometres per hour and time is in seconds, the calculation will be wrong unless velocity is converted first. The safest method is to convert everything into SI units before solving, then convert the final answer if needed.
| Quantity | Common unit | SI conversion |
|---|---|---|
| Velocity | km/h | \(1\,km/h = 0.27778\,m/s\) |
| Velocity | mph | \(1\,mph \approx 0.44704\,m/s\) |
| Distance | ft | \(1\,ft = 0.3048\,m\) |
| Distance | mile | \(1\,mi \approx 1609.344\,m\) |
| Time | minute | \(1\,min = 60\,s\) |
Step-by-Step Strategy
- Read the problem carefully and identify whether acceleration is constant.
- Choose a positive direction.
- Write down the known values with signs and units.
- Write down the unknown variable.
- Select the equation that contains the knowns and the unknown.
- Convert all values to consistent units.
- Substitute values into the formula.
- Solve algebraically before rounding.
- Check whether the answer has the right sign and unit.
- Ask whether the result makes physical sense.
Worked Example 1: Car Accelerating From Rest
A car starts from rest and accelerates at \(3\,m/s^2\) for \(10\,s\). Find its final velocity and displacement.
The car reaches \(30\,m/s\) and travels \(150\,m\). The displacement is not \(30 \times 10 = 300\,m\) because the car was not moving at \(30\,m/s\) for the whole time. It started at rest and gradually increased speed.
Worked Example 2: Dropped Object
An object is dropped from a height of \(80\,m\). Ignoring air resistance, find the time to hit the ground.
The time is about \(4.04\,s\). The impact speed can be found from \(v=\sqrt{2gh}\), which gives about \(39.6\,m/s\).
Worked Example 3: Stopping Distance
A car moving at \(20\,m/s\) brakes with acceleration \(-5\,m/s^2\). Find the stopping distance.
The car travels \(40\,m\) before stopping. In real life, total stopping distance also includes reaction distance, road conditions, tyre condition, slope, and braking system performance.
Common Mistakes
- Using the equations when acceleration is not constant.
- Mixing kilometres per hour with seconds without converting.
- Forgetting that displacement can be negative.
- Using distance when the problem requires displacement.
- Using \(g=+9.8\) when upward was chosen as positive.
- Rounding too early and losing accuracy.
- Choosing an equation that contains an unknown not given in the problem.
- Assuming projectile horizontal velocity changes without air resistance.
Why Kinematic Equations Matter
Kinematic equations are not just school formulas. They are the foundation of motion analysis in engineering, sports science, robotics, transportation, animation, game physics, accident reconstruction, aerospace, and biomechanics. Engineers use related motion models to estimate braking distances, runway lengths, elevator motion, machine movement, and robot trajectories. Game developers use motion equations to animate jumping, falling, and collisions. Coaches use velocity and acceleration ideas to analyze sprinting, throwing, jumping, and projectile paths.
In advanced physics, kinematics becomes more general. Instead of constant acceleration equations, students use calculus:
These definitions work even when acceleration changes. The familiar kinematic equations are special cases that come from integrating constant acceleration. Learning them well builds the mental structure needed for calculus-based physics.
Frequently Asked Questions
Are kinematic equations only for straight-line motion?
The basic SUVAT equations are usually introduced for one-dimensional straight-line motion. Projectile motion uses the same equations separately in horizontal and vertical directions. More advanced vector kinematics extends the same ideas into two and three dimensions.
What does constant acceleration mean?
Constant acceleration means the velocity changes by equal amounts in equal time intervals. If acceleration is \(2\,m/s^2\), velocity changes by \(2\,m/s\) every second.
Can acceleration be negative?
Yes. Negative acceleration means acceleration points in the negative direction. It does not always mean slowing down. If an object is moving in the negative direction and has negative acceleration, its speed may increase.
Is displacement the same as distance?
No. Distance is total path length. Displacement is change in position with direction. Kinematic equations normally use displacement.
Why is the projectile path curved?
Horizontal velocity is constant while vertical velocity changes due to gravity. This combination creates a parabolic trajectory in ideal projectile motion.
Can these equations be used with air resistance?
Not directly. Air resistance usually creates acceleration that changes with speed. For introductory problems, air resistance is often ignored. For real-world modeling, drag equations or numerical simulation may be needed.
Which equation should I use first?
List the variables you know and the variable you need. Choose the equation that contains those variables and avoids unnecessary unknowns.
