How to Calculate Kinetic Energy, Potential Energy, and Work

KE = 0.5 x m x v^2 (m = mass in kg, v = velocity in m/s, KE in joules) Car at motorway speed (110 km/h = 30.6 m/s), mass 1,400 kg: KE = 0.5 x 1,400 x 30.6^2 = 700 x 936.4 = 655,480 J = 655.5 kJ Same car at 55 km/h (15.3 m/s): KE = 0.5 x 1,400 x 15.3^2 = 700 x 234.1 = 163,870 J Key insight: doubling speed quadruples kinetic energy. This is why stopping distance increases as the SQUARE of speed, not linearly — and why speed kills.

GPE = m x g x h (m = mass kg, g = 9.81 m/s^2, h = height in metres) Person (70kg) climbing a 200m hill: GPE = 70 x 9.81 x 200 = 137,340 J = 137.3 kJ This represents the minimum energy needed to climb the hill. Real energy cost is higher because human muscles are only ~25% efficient. Actual calorie cost: 137,340 J / (4,184 J/kcal x 0.25 efficiency) = 131 kcal

Work = Force x Distance x cos(angle) (angle = angle between force and direction of motion) Pushing a trolley horizontally with 80N force over 50m: Work = 80 x 50 x cos(0°) = 80 x 50 x 1 = 4,000 J Pushing at 30° angle to horizontal (handles angled down): Work = 80 x 50 x cos(30°) = 80 x 50 x 0.866 = 3,464 J (Less work done in direction of motion — you're partly pushing into the floor) If force is perpendicular to motion (cos 90° = 0): Work = 0 — no energy transferred in direction of motion (Gravity does zero work on a horizontal surface — makes sense)

Total mechanical energy = KE + GPE = constant (no friction) Ball dropped from 20m height (mass 0.5kg): At top: GPE = 0.5 x 9.81 x 20 = 98.1 J, KE = 0 At bottom: KE = 98.1 J (all PE converted) Velocity: KE = 0.5mv^2 → 98.1 = 0.5 x 0.5 x v^2 → v = 19.8 m/s Roller coaster: if the first drop is 40m: GPE lost = 0.5 x m x 9.81 x 40 = 196.2m J KE at bottom: 196.2m J → v = sqrt(2 x 9.81 x 40) = 28.0 m/s = 100.8 km/h (This gives the theoretical speed; real speed is lower due to friction)