Kinematics with real examples: uniform and accelerated motion

Rectilinear motion is the first model studied in kinematics because it is the simplest one that still describes reality well. With two scenarios — constant velocity and constant acceleration — you can solve most problems in a physics course and a good share of everyday situations: braking, falling, overtaking.

URM: constant velocity, zero acceleration

In URM, velocity is constant and acceleration is exactly zero. The only formula needed is d = v × t. The position-time graph is a straight line with slope equal to velocity; the velocity-time graph is a horizontal line. Real-world approximations: a train on autopilot at cruising speed, a skate gliding on frictionless ice.

UARM: constant non-zero acceleration

In UARM, acceleration is constant (but ≠ 0), causing velocity to change uniformly. The kinematic equations are: v = v₀ + a×t (final velocity), d = v₀×t + ½×a×t² (position), v² = v₀² + 2×a×d (without time). The best-known example: free fall with acceleration g ≈ 9.8 m/s².

Solved URM exercise

A car travels at 90 km/h (= 25 m/s). How long to cover 500 m? t = d/v = 500/25 = 20 s. What distance does it cover in 3 minutes (180 s)? d = v × t = 25 × 180 = 4,500 m = 4.5 km.

Solved UARM exercise

A car starts from rest (v₀ = 0) with acceleration a = 2 m/s². What speed does it reach after 10 s? v = 0 + 2 × 10 = 20 m/s = 72 km/h. What distance did it cover? d = 0 × 10 + ½ × 2 × 10² = 100 m. What acceleration is needed to go from 0 to 100 km/h (27.8 m/s) in 8 s? a = 27.8/8 = 3.47 m/s².

Which equation to choose

The commonest error is not in the arithmetic but in picking the formula. The practical rule is to look at which value the question does not give you. If time is absent, the equation is v² = v₀² + 2·a·d, which dispenses with it entirely. If you do not know the final velocity, use d = v₀·t + ½·a·t². And if the distance is unknown, v = v₀ + a·t. Spotting the missing quantity settles the choice almost every time.

Why braking distance grows so fast with speed

In the equation v² = v₀² + 2·a·d, velocity appears squared. That means the distance needed to stop does not grow in proportion to speed but to its square: doubling the speed quadruples the braking distance. A car that stops in 20 metres from 50 km/h needs around 80 metres from 100 km/h, with identical braking.

On top of that you must add the ground covered during reaction time, which is proportional to speed. This is the physical reason speed limits are not arbitrary, and why an apparently small excess radically changes the outcome of an unexpected event.

Accelerated motion in detail: when velocity changes

Uniformly Accelerated Rectilinear Motion (MRUA in Spanish) describes an object moving in a straight line with constant acceleration. Unlike uniform motion (zero acceleration), here the velocity changes at a fixed rate. It is one of the most important models in classical physics because it describes everyday situations: a car accelerating from traffic lights, a stone falling through the air, or an aircraft rolling down the runway before take-off.

The 4 kinematic equations of MRUA

• •v = v₀ + a·t → final velocity as a function of time
• •x = v₀·t + ½·a·t² → position (displacement) as a function of time
• •v² = v₀² + 2·a·x → velocity without needing to know time
• •x = ½·(v₀ + v)·t → displacement as average velocity × time

Variables and units

v₀ is initial velocity (m/s), v is final velocity (m/s), a is acceleration (m/s²), t is time (s) and x is displacement (m). Acceleration can be positive (object speeds up) or negative (object slows down, braking). Earth's gravitational acceleration is a = −9.8 m/s² when the positive axis points upward.

Example: free fall from 45 m

An object falls from 45 m with no initial velocity. a = 9.8 m/s², v₀ = 0. Fall time: x = ½·a·t² → 45 = ½·9.8·t² → t = √(90/9.8) ≈ 3.03 s. Impact velocity: v = v₀ + a·t = 0 + 9.8·3.03 ≈ 29.7 m/s (≈ 107 km/h). Verification with the 3rd equation: v² = 0 + 2·9.8·45 = 882 → v ≈ 29.7 m/s. ✓

Difference between MRUA and MRU

In MRU (Uniform Rectilinear Motion) acceleration is zero: velocity does not change and position grows linearly with time (x = v·t). In MRUA acceleration is non-zero: velocity changes linearly with time and position grows parabolically (x = v₀·t + ½·a·t²). The v-t graph of MRU is a horizontal line; that of MRUA is an inclined line.