The Physics Formulas Behind Motion, Energy, and Force
Newton's second law, kinetic energy, and projectile motion explained with real numbers you can check by hand.
Classical mechanics is built from a small number of relationships that, once memorized, explain an enormous range of everyday physical phenomena.
Force: Newton's second law
F = ma is arguably the most famous equation in physics after E=mc². A 10kg object accelerating at 9.8 m/s² (roughly Earth's gravity) experiences a force of F = 10 × 9.8 = 98 newtons — which is also, not coincidentally, that object's weight due to gravity.
Kinetic energy: why speed matters more than mass
Kinetic energy is KE = ½mv² — notice velocity is squared, but mass isn't. Double an object's mass and its kinetic energy doubles; double its velocity and kinetic energy quadruples. A 5kg object moving at 10 m/s carries KE = 0.5 × 5 × 10² = 250 joules; if it instead moved at 20 m/s (still 5kg), it would carry 4x that energy — 1,000 joules — which is exactly why small increases in speed matter so much for impact severity in vehicle collisions.
Potential energy and the energy trade-off
Gravitational potential energy is PE = mgh. As an object falls, potential energy converts into kinetic energy at the same rate it's lost — a 5kg object at 10m height has PE = 5 × 9.8 × 10 = 490 joules; right before hitting the ground (ignoring air resistance), nearly all of that has converted into kinetic energy, letting you solve for its impact velocity without ever calculating acceleration directly.
Projectile motion: why 45 degrees is special
A projectile's horizontal range follows Range = v²sin(2θ)/g. Since sin(2θ) is maximized when 2θ = 90° (i.e., θ = 45°), a 45-degree launch angle produces the maximum range for any given launch speed, ignoring air resistance. Launch something at 25 m/s at 45°: Range = 25² × sin(90°) / 9.8 = 625 × 1 / 9.8 ≈ 63.8 meters. Any other angle at the same speed travels a shorter horizontal distance.
Momentum and force over time
Momentum (p = mv) is conserved in collisions — the total momentum before a collision equals the total after, which is the entire basis for analyzing car crashes, billiard balls, and rocket propulsion without needing to know the messy details of the collision itself.
Calculate it yourself
- The force calculator, kinetic energy calculator, and potential energy calculator cover the core energy/force relationships
- The projectile range calculator shows range, max height, and time of flight together
- The momentum calculator and centripetal force calculator extend into circular and collision motion