Compute kinetic energy (½·m·v²), linear momentum, and visualize the quadratic energy–velocity relationship. Supports mass units (kg, g, lb) and velocity units (m/s, km/h, mph).
In classical mechanics, the kinetic energy of an object is the energy it possesses due to motion, defined as KE = ½·m·v² where m is mass and v is speed relative to a reference frame. This fundamental equation emerges from the work‑energy theorem: the net work done on a rigid body equals its change in kinetic energy. The quadratic dependence on velocity explains why high‑speed collisions release enormous energy — a car crashing at 100 km/h has four times the destructive energy compared to 50 km/h.
$$ KE = \frac{1}{2} m v^2 \quad \text{(Joules)} \qquad p = m v \quad \text{(kg·m/s)} $$
The concept was refined by Gottfried Leibniz (vis viva) and later formalized by Émilie du Châtelet and Gaspard-Gustave Coriolis. Together with momentum, kinetic energy forms the core of collision analysis — elastic vs. inelastic collisions, ballistic pendulums, and modern airbag engineering.
1. Convert mass to kilograms (kg): 1 g = 0.001 kg, 1 lb = 0.45359237 kg.
2. Convert velocity to meters per second (m/s): 1 km/h = 0.27777778 m/s, 1 mph = 0.44704 m/s.
3. Apply KE = 0.5 × m_kg × (v_m/s)² → result in Joules.
4. Momentum p = m_kg × v_m/s (kg·m/s).
5. Additional conversions: calories (1 cal = 4.184 J), electronvolts (1 eV = 1.602176634e-19 J).
A typical sedan with mass 1500 kg traveling at 90 km/h (25 m/s) stores KE = 0.5×1500×(25)² = 468,750 J. That energy must be dissipated by the car’s crumple zone and braking system. For a child weighing 20 kg, the same speed yields KE = 6,250 J — showing why seatbelts and airbags are critical. The interactive tool allows you to compare different masses and velocities instantly, giving insight for forensic engineering and vehicle design.
The work-energy theorem states that W_net = ΔKE. This principle governs roller coaster design, rocket propulsion, and even particle accelerators. Our graph visualizes how KE changes with velocity for a fixed mass — the slope d(KE)/dv = m·v (momentum) — linking both quantities. Additionally, the calculator computes momentum, essential for impulse (Δp = F·Δt) estimation.
| Object / Scenario | Mass (kg) | Velocity (m/s) | KE (kJ) | Momentum (kg·m/s) |
|---|---|---|---|---|
| Bowling ball | 7.2 | 8 | 0.23 | 57.6 |
| Running sprinter | 80 | 10 | 4.0 | 800 |
| Small airplane | 1200 | 60 | 2160 | 72000 |
| High‑speed train | 40000 | 45 | 40500 | 1.8e6 |
Emilie du Châtelet (1706‑1749) was among the first to derive the correct formula KE ∝ mv², building on Leibniz's vis viva (mv²). Later, James Prescott Joule established the mechanical equivalent of heat, linking kinetic energy to thermal energy. Modern resources: NIST, Halliday & Resnick "Fundamentals of Physics", and the Feynman Lectures.