MRUA: what is uniformly accelerated motion and how to calculate it

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.