Derivation & Meaning of Equations
Kinematics is the branch of physics that describes how objects move, without asking why they move. When an object accelerates uniformly — meaning its velocity changes by the same amount every second — we can describe its entire motion using just four variables: initial velocity, final velocity, acceleration, and time (or displacement). The kinematic equations are a set of four algebraic relationships connecting these variables. They are not arbitrary formulae to memorise blindly; each one can be derived from first principles using simple algebra and geometry, and each carries a precise physical meaning. The central assumption underlying all kinematic equations is that acceleration is constant throughout the motion. This is a powerful simplification: it means the velocity-time graph is a straight line, and all the equations flow naturally from that single geometric fact. In reality, many everyday situations approximate this well — a car braking hard, a ball in free fall near Earth's surface, or a sprinter during the first second of a race. However, these equations break down the moment acceleration varies, so understanding the assumption is just as important as knowing the equations themselves. Galileo Galilei first established that freely falling objects accelerate uniformly, demolishing the Aristotelian idea that heavier objects fall faster. His inclined-plane experiments in the early 17th century laid the experimental groundwork, while later mathematicians formalised his findings into the algebraic forms we use today. Mastering these derivations means you truly understand motion — not just how to plug numbers into formulae.
- Start with the definition of uniform acceleration
- Use the velocity-time graph to find displacement
- Eliminate final velocity to get the third equation
- Eliminate time to get the fourth equation
- Summary of assumptions and limitations
Start with the definition of uniform acceleration
Acceleration is defined as the rate of change of velocity: a = (v − u) / t, where u is initial velocity, v is final velocity, a is constant acceleration, and t is time elapsed. Rearranging immediately gives the first kinematic equation: v = u + at. This is simply the definition of constant acceleration written in a more useful form. Physically, it tells you that if you start at speed u and gain speed at rate a for time t, your final speed is v.
Use the velocity-time graph to find displacement
Because acceleration is constant, the v–t graph is a straight line from u to v. Displacement equals the area under this graph — a trapezium with parallel sides u and v and width t. Area = ½(u + v)t, giving the second equation: s = ½(u + v)t. The average velocity of an object with constant acceleration is simply the arithmetic mean of its initial and final velocities, so multiplying by time gives displacement. This geometric insight is elegant: area under a v–t graph always equals displacement, regardless of the shape.
Eliminate final velocity to get the third equation
Substitute v = u + at into s = ½(u + v)t: s = ½(u + u + at)t = ½(2u + at)t, which simplifies to s = ut + ½at². This equation is perhaps the most famous kinematic result. The term ut is the displacement the object would have if it continued at constant speed u; the term ½at² is the extra displacement gained due to acceleration. Note that if a = 0, the equation reduces to s = ut — uniform motion, as expected.
Eliminate time to get the fourth equation
From v = u + at, we get t = (v − u)/a. Substitute into s = ½(u + v)t: s = ½(u + v)(v − u)/a = (v² − u²)/(2a). Rearranging: v² = u² + 2as. This equation is invaluable when time is not known or not wanted. It connects final velocity directly to displacement and acceleration. For a car braking from 30 m/s to rest, for instance, you can calculate stopping distance without knowing braking time — enormously useful for road-safety calculations.
Summary of assumptions and limitations
All four equations assume: (1) acceleration is constant in both magnitude and direction; (2) motion is along a straight line (one dimension); (3) the values of u, v, a, s, and t are all measured from the same reference point and in consistent sign conventions. If acceleration changes — for example, a rocket whose thrust varies, or a pendulum — these equations cannot be applied directly. In those cases, calculus (integration) is needed. Always check the assumption of uniform acceleration before applying these equations.
The Key Equation
- v — Final velocity (m s⁻¹): The velocity of the object at the end of the interval being considered.
- u — Initial velocity (m s⁻¹): The velocity of the object at the start of the interval being considered.
- a — Acceleration (m s⁻²): The constant rate of change of velocity; positive if speeding up in the chosen positive direction, negative if slowing down.
- s — Displacement (m): The vector distance from the starting position to the finishing position, measured in the direction of motion. Not the same as total distance if the object reverses direction.
Worked Example
Problem. A car is travelling at 20 m/s when the driver applies the brakes, producing a constant deceleration of 4 m/s². How far does the car travel before coming to rest?
Answer. The car travels 50 metres before stopping.
Origins · 1 minGalileo Galilei and how this was discovered
Galileo was the first to demonstrate experimentally that freely falling bodies accelerate uniformly, regardless of their mass. By rolling brass balls down inclined planes and timing them with water clocks, he showed that distance increases as the square of time — precisely the relationship encoded in s = ut + ½at². His work replaced centuries of Aristotelian speculation with quantitative, reproducible measurement, founding the science of kinematics. Because clocks of his era were too imprecise to time free fall directly, Galileo tilted his ramps at shallow angles to 'dilute' gravity, slowing the motion enough to measure accurately — a brilliant experimental workaround that even modern physicists admire.
Related Concepts
- Displacement, Velocity and Acceleration — The kinematic equations are built from the definitions of displacement, velocity and acceleration; understanding these quantities is essential before deriving or using the equations.
- Velocity–Time Graphs — The geometric derivation of kinematic equations relies directly on interpreting the area and gradient of a velocity–time graph for uniform acceleration.
- Free Fall and Acceleration Due to Gravity — Free fall is the most important physical example of uniform acceleration, where a = g ≈ 9.81 m/s², and the kinematic equations are applied directly to solve projectile problems.
- Relative Motion and Frames of Reference — Kinematic equations assume a fixed reference frame; understanding relative motion clarifies when and how to choose initial conditions and sign conventions correctly.