Applying Kinematic Equations
When objects move with a constant acceleration — think of a car pressing steadily on the accelerator, or a ball falling under gravity — their motion can be described completely by four elegant equations. These kinematic equations link five variables: initial velocity, final velocity, acceleration, displacement, and time. Mastering how to apply them is one of the most practically useful skills in physics, letting you predict where a projectile lands, how long a braking car takes to stop, or how fast a skydiver is falling after a given time. The key to solving any kinematics problem is identifying which three of the five variables are known, then selecting the equation that contains those three plus the one unknown you need to find. Think of it like a puzzle: each equation is a different 'window' onto the same motion, and your job is to choose the right window. This systematic approach — list knowns, identify the unknown, choose the equation, rearrange, substitute — works reliably on every uniform-acceleration problem you will ever meet. It is important to remember that these equations only apply when acceleration is constant. In real situations, acceleration often varies, but for many everyday scenarios — free fall near Earth's surface, a car with cruise control engaged on a flat road — the constant-acceleration model is an excellent approximation that yields accurate, testable predictions.
- Define a positive direction and set up a coordinate system
- List the five kinematic variables
- Select the appropriate equation
- Rearrange algebraically, then substitute values
- Check the answer for reasonableness
Define a positive direction and set up a coordinate system
Before writing a single number, decide which direction is positive. For horizontal motion, right is usually positive. For vertical motion, upward or downward can be positive — just be consistent throughout the problem. Label your diagram with an arrow showing the positive direction. This prevents sign errors, which are the most common source of mistakes in kinematics.
List the five kinematic variables
Write down: u (initial velocity), v (final velocity), a (acceleration), s (displacement), and t (time). Fill in everything the problem tells you. Pay careful attention to language — 'starts from rest' means u = 0, 'comes to a stop' means v = 0, 'falls freely' means a = 9.81 m s⁻² downward. Put a question mark next to the variable you are solving for. If you have three knowns and one unknown, you have enough information to proceed.
Select the appropriate equation
There are four kinematic equations. Match them to your known variables: (1) v = u + at — use when s is not needed. (2) s = ut + ½at² — use when v is not needed. (3) v² = u² + 2as — use when t is not needed. (4) s = ½(u + v)t — use when a is not needed. The variable that does NOT appear in your chosen equation is the one you are neither given nor asked for — this is sometimes called the 'missing variable' technique.
Rearrange algebraically, then substitute values
Always rearrange your chosen equation for the unknown before substituting numbers. This reduces arithmetic errors and makes it easier to check your working. Once you substitute, include units in every step — they act as a built-in error check. If your units do not cancel to give the correct unit for your answer (e.g. m s⁻¹ for velocity), something has gone wrong algebraically or with unit conversion.
Check the answer for reasonableness
Ask yourself: does the magnitude make physical sense? A car stopping in 2 mm or 200 km are both suspicious. Check the sign — is the direction consistent with your setup? For two-part problems (e.g. upward then downward flight), treat each phase separately with its own list of variables, carrying the final state of one phase as the initial state of the next.
The Key Equation
- v — Final velocity (m s⁻¹): The velocity of the object at the end of the time interval being considered.
- u — Initial velocity (m s⁻¹): The velocity of the object at the start of the time interval.
- a — Acceleration (m s⁻²): The constant rate of change of velocity. Positive if speeding up in the positive direction, negative if decelerating.
- s — Displacement (m): The change in position of the object, measured in the positive direction. Can be negative if the object moves in the negative direction.
Worked Example
Problem. A cyclist is travelling at 12 m s⁻¹ when she applies her brakes, producing a constant deceleration of 3.0 m s⁻². How far does she travel before coming to a complete stop?
Answer. The cyclist travels 24 metres before stopping.
Origins · 1 minGalileo Galilei and how this was discovered
Galileo was the first scientist to systematically study uniform acceleration through careful experiment. By rolling balls down inclined planes and timing their motion with a water clock, he established that displacement is proportional to the square of time — the mathematical heart of the kinematic equations. He also recognised that, in the absence of air resistance, all objects fall with the same constant acceleration regardless of mass, directly contradicting Aristotle's long-accepted claim. Galileo measured time using his own pulse and the flow of water through a small hole, yet still obtained results accurate enough to form the foundation of classical mechanics. He never had access to a stopwatch.
Related Concepts
- Displacement, Velocity and Acceleration — Understanding the definitions of displacement, velocity, and acceleration is essential before you can correctly identify and substitute values into kinematic equations.
- Deriving the Kinematic Equations — Seeing how the four equations are derived from the definitions of velocity and acceleration helps you understand their limitations and know when they can legitimately be applied.
- Free Fall and Acceleration Due to Gravity — Free fall is the most common real-world application of the kinematic equations, where acceleration is the constant g = 9.81 m s⁻² directed downward.
- Velocity–Time Graphs — Velocity–time graphs provide a visual representation of the same motion described by kinematic equations, and the area under the graph corresponds to displacement — linking graphical and algebraic approaches.