URM and UARM: Differences, Formulas and Solved Examples
thecalculator.tech
22 June 2026 · 5 min read
We explain the differences between uniform rectilinear motion and uniformly accelerated motion, with kinematic formulas and step-by-step solved exercises.
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.
These formulas assume constant acceleration in a straight line without friction. In real situations, other forces come into play.
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