{"id":755,"date":"2026-08-10T02:06:07","date_gmt":"2026-08-10T02:06:07","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=755"},"modified":"2026-08-24T13:03:48","modified_gmt":"2026-08-24T13:03:48","slug":"mechanics-formulas","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/mechanics-formulas\/","title":{"rendered":"Mechanics Formulas Cheat Sheet"},"content":{"rendered":"\n<div class=\"pf-citation\"><div class=\"eyebrow\">Definition<\/div><p>\n \nMechanics formulas are the equations that link force, mass, motion, energy and momentum, letting you predict how an object moves. The core set is small: Newton&#8217;s second law (force equals mass times acceleration), the four constant-acceleration equations, kinetic and potential energy, and momentum as mass times velocity. Everything else in mechanics is built from these.\n \n<\/p><\/div>\n \n<p>Open any mechanics textbook to the back cover and you will find the same thing: a wall of equations, printed small, with no clue about which one to reach for. Most students copy it out, memorise it, and still stall on the first line of a problem.<\/p>\n \n<p>That stall is almost never a memory failure. It is a matching failure \u2014 you know the equations, you just cannot tell which one fits the numbers in front of you. This sheet is arranged to fix that: every formula is grouped by the quantity it produces, every symbol carries its SI unit, and every row that has a calculator on this site links straight to it.<\/p>\n \n<h2>What Are Mechanics Formulas?<\/h2>\n \n<p>Mechanics formulas are the equations describing how forces make objects move, deform, spin and collide. They cover everything from a dropped key to a satellite holding orbit, and they all rest on a handful of conserved quantities and one law relating force to acceleration.<\/p>\n \n<p>Think of them less as a list to store and more as a set of translation rules. Each formula converts a group of measurable things \u2014 a mass, a distance, a time \u2014 into something you cannot measure directly, like the force in a cable or the speed at the bottom of a slope.<\/p>\n \n<p>Mechanics splits neatly into six families, and that split is the whole trick to using the sheet below:<\/p>\n \n<ul>\n<li><strong>Kinematics<\/strong> \u2014 motion described, with no mention of what caused it.<\/li>\n<li><strong>Forces and Newton&#8217;s laws<\/strong> \u2014 what makes motion change.<\/li>\n<li><strong>Work, energy and power<\/strong> \u2014 the bookkeeping of what a force costs.<\/li>\n<li><strong>Momentum and collisions<\/strong> \u2014 what survives an impact.<\/li>\n<li><strong>Circular motion and rotation<\/strong> \u2014 the same laws, rewritten for spin.<\/li>\n<li><strong>Gravitation and oscillations<\/strong> \u2014 orbits, pendulums and springs.<\/li>\n<\/ul>\n \n<h3>Every Symbol Is a Quantity, and Every Quantity Has a Unit<\/h3>\n \n<p>A formula is only half the information. The other half is what each letter stands for and what unit it must be in \u2014 and that second half is where marks are actually lost.<\/p>\n \n<p>Mechanics runs on three SI base units and a small set of derived ones built from them. The <a href=\"https:\/\/www.nist.gov\/pml\/owm\/metric-si\/si-units\" target=\"_blank\" rel=\"noopener\">NIST guide to the SI<\/a> sets out how those derived units are constructed; here are the four reference values that appear again and again below.<\/p>\n \n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr style=\"background:#0A1628;color:#FAF6EE;\">\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Reference value<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Symbol<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Value<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Where it turns up<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Acceleration of free fall (Earth)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">g<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">9.81 m\/s<sup>2<\/sup> (standard value 9.80665 m\/s<sup>2<\/sup>)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Weight, free fall, projectiles, pendulums<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Gravitational constant<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">G<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">6.674 &times; 10<sup>-11<\/sup> N&middot;m<sup>2<\/sup>\/kg<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Universal gravitation, orbits, escape speed<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">The newton<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">N<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">1 N = 1 kg&middot;m\/s<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Every force equation<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">The joule<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">J<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">1 J = 1 N&middot;m = 1 kg&middot;m<sup>2<\/sup>\/s<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Work, kinetic energy, potential energy<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n<p>Notice that the newton and the joule are not new ideas \u2014 they are shorthand for combinations of kilograms, metres and seconds. That fact is what makes the unit check in the section below work at all. If you want the full picture of base and derived units, we cover it in the guide to <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/si-units-physics\/\">SI units in physics<\/a>.<\/p>\n \n<h2>The Mechanics Formulas Cheat Sheet: 40 Equations in 6 Families<\/h2>\n \n<p>Below are the 40 mechanics formulas that cover essentially every problem set at school and first-year university level. Each table gives the quantity you are solving for, the equation, every symbol with its SI unit, and a link to the calculator that does the arithmetic.<\/p>\n \n<p>Scan the left-hand column first. Find the quantity you want, and the row hands you the formula.<\/p>\n \n<h3>1. Kinematics: Motion Without Forces<\/h3>\n \n<p>Kinematics describes motion using only position, velocity, acceleration and time \u2014 it never asks what caused the motion. The four constant-acceleration equations are the workhorses here.<\/p>\n \n<div class=\"pf-formula\">v = u + at<\/div>\n \n<div class=\"pf-formula\">s = ut + \u00bdat<sup>2<\/sup><\/div>\n \n<div class=\"pf-formula\">v<sup>2<\/sup> = u<sup>2<\/sup> + 2as<\/div>\n \n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr style=\"background:#0A1628;color:#FAF6EE;\">\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">You want<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Formula<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Symbols and SI units<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Calculator<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Average velocity<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">v = \u0394s \/ \u0394t<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">v = velocity (m\/s); \u0394s = displacement (m); \u0394t = time (s)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/velocity\">Velocity<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Average acceleration<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">a = (v &minus; u) \/ t<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">a = acceleration (m\/s<sup>2<\/sup>); u = initial velocity (m\/s); v = final velocity (m\/s); t = time (s)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/acceleration\">Acceleration<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Final velocity (no distance given)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">v = u + at<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">as above<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Displacement (no final velocity)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">s = ut + \u00bdat<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">s = displacement (m)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Final velocity (no time given)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">v<sup>2<\/sup> = u<sup>2<\/sup> + 2as<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">all five symbols in m, s and m\/s<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/suvat\">SUVAT<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Displacement (no acceleration given)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">s = \u00bd(u + v)t<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">as above<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Free-fall drop and speed<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">h = \u00bdgt<sup>2<\/sup>, v = gt<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">h = drop height (m); g = 9.81 m\/s<sup>2<\/sup>; t = time (s)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/free-fall\">Free Fall<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Projectile range on level ground<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">R = v<sub>0<\/sub><sup>2<\/sup>&middot;sin(2\u03b8) \/ g<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">R = range (m); v<sub>0<\/sub> = launch speed (m\/s); \u03b8 = launch angle (degrees or radians)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/projectile-motion\">Projectile Motion<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Resultant of perpendicular components<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">R = sqrt(R<sub>x<\/sub><sup>2<\/sup> + R<sub>y<\/sub><sup>2<\/sup>)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">R = resultant magnitude (same unit as components); direction \u03c6 = atan2(R<sub>y<\/sub>, R<sub>x<\/sub>)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/vector-addition\">Vector Addition<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n<p>Each of the four constant-acceleration equations leaves out exactly one of the five quantities. That is not decoration \u2014 it is the selection rule. Pick the equation missing the thing you neither know nor want, and we unpack why in the guide to the <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/kinematics\/suvat-equations\/\">SUVAT equations<\/a>.<\/p>\n \n<h3>2. Forces and Newton&#8217;s Laws<\/h3>\n \n<p>Force equations tell you how motion changes. Newton&#8217;s second law is the hinge that every other force formula swings on, and the rest of this family is really just different ways of working out what goes into <em>F<\/em>.<\/p>\n \n<div class=\"pf-formula\">F = m\u00b7a<\/div>\n \n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr style=\"background:#0A1628;color:#FAF6EE;\">\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">You want<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Formula<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Symbols and SI units<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Calculator<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Net force<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F = m&middot;a<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F = net force (N); m = mass (kg); a = acceleration (m\/s<sup>2<\/sup>)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/newtons-second-law\">Newton&#8217;s Second Law<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Weight<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">W = m&middot;g<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">W = weight (N); m = mass (kg); g = local gravitational field strength (N\/kg, numerically equal to m\/s<sup>2<\/sup>)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/weight-on-other-planets\">Weight on Other Planets<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Friction force<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F<sub>f<\/sub> = \u03bc&middot;N<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F<sub>f<\/sub> = friction force (N); \u03bc = coefficient of friction (no unit); N = normal force (N)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/friction\">Friction<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Acceleration down a rough slope<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">a = g(sin\u03b8 &minus; \u03bc&middot;cos\u03b8)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03b8 = slope angle; \u03bc = coefficient of friction; a in m\/s<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/inclined-plane\">Inclined Plane<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Spring force<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F = &minus;k&middot;x<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">k = spring constant (N\/m); x = extension from natural length (m); the minus sign means the force opposes the stretch<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/hookes-law\">Hooke&#8217;s Law<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Spring constant from a test load<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">k = F \/ x<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">k in N\/m; F in N; x in m<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/spring-constant\">Spring Constant<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Gravitational attraction between two masses<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F = G&middot;m<sub>1<\/sub>&middot;m<sub>2<\/sub> \/ r<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">G = 6.674 &times; 10<sup>-11<\/sup> N&middot;m<sup>2<\/sup>\/kg<sup>2<\/sup>; m<sub>1<\/sub>, m<sub>2<\/sub> in kg; r = centre-to-centre separation (m)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/gravitational-force\">Gravitational Force<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n<p>One detail worth pinning down: the <em>F<\/em> in F = ma is always the <strong>net<\/strong> force \u2014 the vector sum of everything acting. Draw the free-body diagram before you write the equation, a habit set out in full in <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/newtons-laws-of-motion\/\">Newton&#8217;s laws of motion<\/a>.<\/p>\n \n<h3>3. Work, Energy and Power<\/h3>\n \n<p>Energy formulas let you skip the details of the journey. If a question gives you heights, speeds and distances but never mentions time, this family is almost always the shortcut.<\/p>\n \n<div class=\"pf-formula\">KE = \u00bd\u00b7m\u00b7v<sup>2<\/sup><\/div>\n \n<div class=\"pf-formula\">PE = m\u00b7g\u00b7h<\/div>\n \n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr style=\"background:#0A1628;color:#FAF6EE;\">\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">You want<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Formula<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Symbols and SI units<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Calculator<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Work done by a force<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">W = F&middot;d&middot;cos\u03b8<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">W = work (J); F = force (N); d = displacement (m); \u03b8 = angle between force and displacement<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/work-power\">Work &amp; Power<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Kinetic energy<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">KE = \u00bd&middot;m&middot;v<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">KE in J; m in kg; v in m\/s<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/kinetic-energy\">Kinetic Energy<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Gravitational potential energy near Earth<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">PE = m&middot;g&middot;h<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">PE in J; h = height above your chosen zero (m)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/gravitational-potential-energy\">Gravitational Potential Energy<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Elastic potential energy in a spring<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">E<sub>e<\/sub> = \u00bd&middot;k&middot;x<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">E<sub>e<\/sub> in J; k in N\/m; x in m<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Power<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">P = W \/ t = F&middot;v<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">P = power (W); W = work (J); t = time (s); v = speed (m\/s)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/power\">Power<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Mechanical advantage of a machine<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">MA = F<sub>load<\/sub> \/ F<sub>effort<\/sub><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">both forces in N; MA has no unit<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/mechanical-advantage\">Mechanical Advantage<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n<p>The joule doing the work here is the same joule that heats a kettle. That single currency running across mechanics, heat and electricity is the point of the whole idea, explored further in <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/what-is-energy-in-physics\/\">what energy is in physics<\/a>.<\/p>\n \n<h3>4. Momentum and Collisions<\/h3>\n \n<p>Momentum formulas govern what happens when things hit each other. Their power is that total momentum is conserved in every collision \u2014 even the messy ones where energy is clearly lost to noise, heat and crumpled metal.<\/p>\n \n<div class=\"pf-formula\">p = m\u00b7v<\/div>\n \n<div class=\"pf-formula\">J = F\u00b7\u0394t = \u0394p<\/div>\n \n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr style=\"background:#0A1628;color:#FAF6EE;\">\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">You want<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Formula<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Symbols and SI units<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Calculator<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Momentum<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">p = m&middot;v<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">p = momentum (kg&middot;m\/s); m in kg; v in m\/s. Momentum is a vector \u2014 direction matters<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/momentum\">Momentum<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Impulse<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">J = F&middot;\u0394t = \u0394p<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">J = impulse (N&middot;s, identical to kg&middot;m\/s); F = average force (N); \u0394t = contact time (s)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/impulse\">Impulse<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Velocities after a collision<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">m<sub>1<\/sub>u<sub>1<\/sub> + m<sub>2<\/sub>u<sub>2<\/sub> = m<sub>1<\/sub>v<sub>1<\/sub> + m<sub>2<\/sub>v<sub>2<\/sub><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">masses in kg; u = before, v = after, both in m\/s, signed by direction<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/collision\">Collision<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Centre of mass of several bodies<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">x<sub>cm<\/sub> = \u03a3(m<sub>i<\/sub>&middot;x<sub>i<\/sub>) \/ \u03a3m<sub>i<\/sub><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">x<sub>cm<\/sub> = position (m); m<sub>i<\/sub> = each mass (kg); x<sub>i<\/sub> = each position (m)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/center-of-mass\">Center of Mass<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n<p>A quick sanity check students find useful: N&middot;s and kg&middot;m\/s are the same unit written two ways. If your impulse and your momentum change do not come out numerically equal, one of them has an arithmetic error.<\/p>\n \n<h3>5. Circular Motion and Rotation<\/h3>\n \n<p>Rotational formulas are the linear ones in disguise. Swap distance for angle, mass for moment of inertia and force for torque, and the shapes of the equations repeat almost line for line.<\/p>\n \n<div class=\"pf-formula\">F_c = m\u00b7v<sup>2<\/sup> \/ r<\/div>\n \n<div class=\"pf-formula\">\u03c4 = r\u00b7F\u00b7sin\u03b8<\/div>\n \n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr style=\"background:#0A1628;color:#FAF6EE;\">\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">You want<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Formula<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Symbols and SI units<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Calculator<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Centripetal acceleration<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">a<sub>c<\/sub> = v<sup>2<\/sup> \/ r<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">a<sub>c<\/sub> in m\/s<sup>2<\/sup>; v = tangential speed (m\/s); r = radius (m)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/circular-motion\">Circular Motion<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Centripetal force<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F<sub>c<\/sub> = m&middot;v<sup>2<\/sup> \/ r = m&middot;\u03c9<sup>2<\/sup>&middot;r<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">F<sub>c<\/sub> in N; \u03c9 = angular velocity (rad\/s)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/centripetal-force\">Centripetal Force<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Angular velocity (and speed from it)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03c9 = \u03b8 \/ t = 2\u03c0 \/ T, and v = \u03c9&middot;r<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03c9 in rad\/s; \u03b8 in radians; T = period (s); v in m\/s<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/angular-velocity\">Angular Velocity<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Torque<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03c4 = r&middot;F&middot;sin\u03b8<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03c4 = torque (N&middot;m); r = distance from pivot (m); \u03b8 = angle between r and F<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/torque\">Torque<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Moment of inertia<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">I = k&middot;m&middot;r<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">I in kg&middot;m<sup>2<\/sup>; k = shape factor (\u00bd for a solid disc, 1 for a hoop, 2\/5 for a solid sphere)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/moment-of-inertia\">Moment of Inertia<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Angular acceleration from torque<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03c4 = I&middot;\u03b1<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03b1 = angular acceleration (rad\/s<sup>2<\/sup>) \u2014 the rotational twin of F = ma<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Rotational kinetic energy<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">KE<sub>rot<\/sub> = \u00bd&middot;I&middot;\u03c9<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">KE<sub>rot<\/sub> in J; I in kg&middot;m<sup>2<\/sup>; \u03c9 in rad\/s<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Angular momentum<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">L = I&middot;\u03c9<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">L in kg&middot;m<sup>2<\/sup>\/s; conserved when no external torque acts<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Ideal banking angle for a curve<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">tan\u03b8 = v<sup>2<\/sup> \/ (r&middot;g)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03b8 = banking angle; v = design speed (m\/s); r = radius (m); g = 9.81 m\/s<sup>2<\/sup><\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/banked-curve\">Banked Curve<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n<p>Angles must be in radians whenever they sit inside \u03c9, \u03b1 or a period. Feed a calculator degrees where it expects radians and the answer comes out wrong by a factor of about 57.<\/p>\n \n<h3>6. Gravitation and Oscillations<\/h3>\n \n<p>These formulas handle the two situations where motion repeats: orbits and oscillations. Both come down to a restoring influence pulling something back towards where it started.<\/p>\n \n<div class=\"pf-formula\">T = 2\u03c0\u00b7sqrt(L \/ g)<\/div>\n \n<div class=\"pf-table-scroll\" style=\"display:block;width:100%;max-width:100%;overflow-x:auto;-webkit-overflow-scrolling:touch;margin:1.5em 0;\">\n<table style=\"width:100%;border-collapse:collapse;\">\n<thead>\n<tr style=\"background:#0A1628;color:#FAF6EE;\">\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">You want<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Formula<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Symbols and SI units<\/th>\n<th style=\"padding:10px;border:1px solid #D9CFB8;text-align:left;\">Calculator<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Escape velocity<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">v<sub>esc<\/sub> = sqrt(2&middot;G&middot;M \/ r)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">v<sub>esc<\/sub> in m\/s; M = mass of the body (kg); r = launch radius from its centre (m)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/escape-velocity\">Escape Velocity<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Orbital period (Kepler&#8217;s third law)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">T<sup>2<\/sup> = 4\u03c0<sup>2<\/sup>&middot;a<sup>3<\/sup> \/ (G&middot;M)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">T = period (s); a = semi-major axis (m); M = central mass (kg)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/kepler-third-law\">Kepler&#8217;s Third Law<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Period of a simple pendulum<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">T = 2\u03c0&middot;sqrt(L \/ g)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">T in s; L = string length (m); valid for small swings, roughly under 15 degrees<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/pendulum-period\">Pendulum Period<\/a><\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Period of a mass on a spring<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">T = 2\u03c0&middot;sqrt(m \/ k)<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">T in s; m in kg; k in N\/m. Note it does not depend on amplitude<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">&mdash;<\/td>\n<\/tr>\n<tr>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">Simple harmonic motion relations<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">\u03c9 = 2\u03c0f, T = 1 \/ f, v<sub>max<\/sub> = \u03c9&middot;A<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\">f = frequency (Hz); A = amplitude (m); v<sub>max<\/sub> in m\/s<\/td>\n<td style=\"padding:10px;border:1px solid #D9CFB8;\"><a href=\"https:\/\/physicsfundamentalsinfo.com\/calculators\/simple-harmonic-motion\">Simple Harmonic Motion<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n \n<h2>How Do You Choose the Right Mechanics Formula?<\/h2>\n \n<p>Choose a mechanics formula by naming the quantity you want, listing the quantities you were given, and then picking the family whose equation contains only those symbols and nothing else. Three steps, in this order:<\/p>\n \n<ol>\n<li><strong>Name the target.<\/strong> Write down the symbol and unit of the one thing the question asks for \u2014 &#8220;v, in m\/s&#8221;, not &#8220;the speed bit&#8221;.<\/li>\n<li><strong>List what you have.<\/strong> Extract every number with its symbol and unit. Do this before you look at any formula.<\/li>\n<li><strong>Match, don&#8217;t hunt.<\/strong> Pick the equation whose symbols are covered by your list plus your target. If one symbol is left over that you neither know nor want, that is the wrong equation.<\/li>\n<\/ol>\n \n<p>Step three is where the sheet above earns its keep. Because the tables are keyed by &#8220;you want&#8221;, you are searching a short column rather than scanning forty equations.<\/p>\n \n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/mechanics-formulas-decision-map-choosing-right-formula.webp\" width=\"1520\" height=\"910\" alt=\"Mechanics formulas - Decision map for choosing the right mechanics formula, routing from the quantity you want to one of four formula families\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:760px;display:block;margin:0 auto;\" \/><\/figure>\n \n<p style=\"text-align:center;font-size:13px;font-style:italic;color:#1F2E47;\">Routing from what a question gives you to the right family of mechanics formulas.<\/p>\n \n<h2>How Do You Rearrange a Mechanics Formula?<\/h2>\n \n<p>Rearrange a mechanics formula by doing the same operation to both sides until the symbol you want stands alone \u2014 always in symbols first, before any number goes in. Substituting early is the single most common source of arithmetic slips.<\/p>\n \n<p>For the many mechanics formulas shaped like A = B \u00d7 C \u2014 F = ma, p = mv, W = Fd \u2014 there is a shortcut worth internalising. Cover the symbol you want: if it was on top you multiply the two below, and if it was on the bottom you divide the top by the other.<\/p>\n \n<p>The lab below lets you try it. Pick a relationship, choose which variable to solve for, and watch both the rearranged equation and the unit check update as you move the sliders.<\/p>\n \n<div class=\"pf-sim-slot\"><div class=\"pf-sim-slot-header\"><span class=\"icon-dot\"><\/span><span class=\"label\">Physics Formula Rearranger Lab<\/span><\/div><div class=\"pf-sim-slot-body\"><style>.pf-sim-frame{width:100%;border:none;height:560px}@media(max-width:760px){.pf-sim-frame{height:840px}}<\/style><iframe src=\"\/labs\/physics-formulas.html?embed=1\" class=\"pf-sim-frame\" loading=\"lazy\"><\/iframe><\/div><\/div>\n \n<p>One warning the lab makes obvious: the triangle only works for the A = B \u00d7 C shape. An equation like v<sup>2<\/sup> = u<sup>2<\/sup> + 2as has four terms and a square in it, so no triangle will save you \u2014 that one has to be rearranged properly, line by line.<\/p>\n \n<h2>How Do You Check a Mechanics Formula With Units?<\/h2>\n \n<p>Check a mechanics formula by replacing every symbol with its SI unit and simplifying: if the right-hand side does not collapse to the unit of the quantity you wanted, the formula is wrong. This takes about ten seconds and catches a whole class of errors that re-reading never will.<\/p>\n \n<p>Take kinetic energy. Substituting units into \u00bdmv<sup>2<\/sup> gives kg \u00d7 (m\/s)<sup>2<\/sup>, which is kg\u00b7m<sup>2<\/sup>\/s<sup>2<\/sup>. Since 1 J = 1 N\u00b7m = 1 kg\u00b7m<sup>2<\/sup>\/s<sup>2<\/sup>, the units land exactly on the joule \u2014 the formula survives.<\/p>\n \n<p>Now try the classic slip of writing kinetic energy as \u00bdmv. That gives kg\u00b7m\/s, which is momentum, not energy. The unit check flags it instantly, without you needing to remember which version was right.<\/p>\n \n<p>Three habits make this reliable in practice:<\/p>\n \n<ul>\n<li><strong>Convert to SI before substituting.<\/strong> Grams to kilograms, centimetres to metres, minutes to seconds \u2014 every time.<\/li>\n<li><strong>Strip the pure numbers.<\/strong> Factors like \u00bd, 2\u03c0 and cos\u03b8 carry no units, so ignore them during the check.<\/li>\n<li><strong>Expand the named units.<\/strong> Rewrite N as kg\u00b7m\/s<sup>2<\/sup> and J as kg\u00b7m<sup>2<\/sup>\/s<sup>2<\/sup>, then cancel.<\/li>\n<\/ul>\n \n<p>A unit check confirms a formula is dimensionally possible; it cannot catch a wrong numerical factor. Both \u00bdmv<sup>2<\/sup> and 2mv<sup>2<\/sup> pass the check, so it is a filter, not a proof. For the wider method that surrounds this step, see our <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/solve-physics-problems\/\">six-step framework for solving physics problems<\/a>.<\/p>\n \n<h2>Where These Mechanics Formulas Show Up in Real Life<\/h2>\n \n<p>Every equation on this sheet was written to describe something someone needed to build or predict. Four examples where they are still doing exactly that:<\/p>\n \n<ul>\n<li><strong>Crumple zones.<\/strong> Impulse says J = F\u0394t. A car body that deforms stretches \u0394t across a longer collision, so the same change in momentum arrives as a smaller average force on the occupants.<\/li>\n<li><strong>Motorway curve design.<\/strong> The banking angle formula tan\u03b8 = v<sup>2<\/sup>\/(rg) sets the camber engineers build into a bend so vehicles at the design speed need no help from friction to turn.<\/li>\n<li><strong>Figure skating spins.<\/strong> Angular momentum L = I\u03c9 is conserved when a skater pulls their arms in. Moment of inertia drops, so angular velocity climbs \u2014 no push required.<\/li>\n<li><strong>Satellite launches.<\/strong> Kepler&#8217;s third law fixes the altitude that gives a 24-hour period, which is why geostationary satellites all sit on the same ring above the equator.<\/li>\n<\/ul>\n \n<h2>Common Mistakes With Mechanics Formulas<\/h2>\n \n<p>The same four errors account for most lost marks in mechanics. Each one is a misunderstanding, not a slip \u2014 which is why they survive re-reading the sheet.<\/p>\n \n<h3>Mistake 1: Treating mass and weight as the same thing<\/h3>\n \n<p>Mass is measured in kilograms and never changes; weight is a force in newtons and changes with location. Writing 60 kg into a formula that wants a force in newtons is one of the most frequent errors on marked scripts.<\/p>\n \n<p>The fix is mechanical: whenever you see a mass in a force equation, ask whether the equation wants m or mg.<\/p>\n \n<h3>Mistake 2: Using F = ma with only one of the forces<\/h3>\n \n<p>The F in Newton&#8217;s second law is the net force, not the force you happen to have been given. A 100 N push on a crate with 30 N of friction gives a net force of 70 N, and using 100 N inflates the acceleration by more than 40 per cent.<\/p>\n \n<h3>Mistake 3: Assuming kinetic energy is conserved in every collision<\/h3>\n \n<p>Momentum is conserved in all collisions; kinetic energy is conserved only in elastic ones. In a real crash a large fraction of the kinetic energy goes into deformation, heat and sound \u2014 so applying the elastic-collision equations to a car pile-up gives a confidently wrong answer.<\/p>\n \n<h3>Mistake 4: Mixing degrees and radians<\/h3>\n \n<p>Angles inside \u03c9, \u03b1, or a period must be in radians; angles inside sin\u03b8 and cos\u03b8 in force resolution are usually in degrees. A pendulum or rotation answer that comes out about 57 times too large or too small almost always traces back to this.<\/p>\n \n<h2>How Mechanics Formulas Connect to the Rest of Physics<\/h2>\n \n<p>Mechanics formulas are not a closed set \u2014 they are the layer everything else is built on top of. The joule you calculate with \u00bdmv<sup>2<\/sup> is the same joule that appears in specific heat capacity; the newton from F = ma is the same newton in electrostatic and magnetic force laws.<\/p>\n \n<p>Within mechanics itself, the families are alternative routes rather than separate subjects. The diagram below shows the same block reaching the same answer two different ways.<\/p>\n \n<figure class=\"pf-figure\" style=\"margin:1.6em 0;\"><img src=\"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-content\/uploads\/2026\/08\/mechanics-formulas-block-incline-solved-force-route.webp\" width=\"1440\" height=\"700\" alt=\"Mechanics formulas - Diagram showing a block on an incline solved by the force route and the energy route, both giving the same final speed\" loading=\"lazy\" decoding=\"async\" style=\"width:100%;height:auto;max-width:720px;display:block;margin:0 auto;\" \/><\/figure>\n \n<p style=\"text-align:center;font-size:13px;font-style:italic;color:#1F2E47;\">Force and energy routes through the same mechanics problem, giving identical answers.<\/p>\n \n<p>Add friction to that slope and the energy route needs an extra term, but it still works. Add a collision, and momentum becomes the only route that survives. Choosing well is what separates a two-line solution from a twenty-line one.<\/p>\n \n<p>If you need equations from outside mechanics \u2014 thermodynamics, waves, electromagnetism or modern physics \u2014 they are all collected in the <a href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/physics-formulas\/\">physics formulas hub<\/a>. MIT&#8217;s introductory mechanics course makes the same argument this page does, and makes it forcefully: its own <a href=\"https:\/\/ocw.mit.edu\/courses\/8-01sc-classical-mechanics-fall-2016\/\" target=\"_blank\" rel=\"noopener\">8.01 Classical Mechanics materials<\/a> open by contrasting learning a set of formulas to memorise with learning principles you can apply to unfamiliar situations.<\/p>\n \n<h2>Worked Problems<\/h2>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 1<\/div><div class=\"pf-problem-question\">A 1,200 kg car accelerates uniformly from rest to 27 m\/s in 9.0 s. Find its acceleration and the net force on it.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Acceleration from the definition, a = (v \u2212 u) \/ t.<\/p>\n<p>Step 2: a = (27 m\/s \u2212 0 m\/s) \/ 9.0 s = 3.0 m\/s<sup>2<\/sup><\/p>\n<p>Step 3: Net force from Newton&#8217;s second law, F = m\u00b7a = 1,200 kg \u00d7 3.0 m\/s<sup>2<\/sup> = 3,600 N<\/p>\n<p><strong>Answer: a = 3.0 m\/s<sup>2<\/sup>, F = 3.6 kN<\/strong><\/p>\n<\/div><\/details><\/div>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 2<\/div><div class=\"pf-problem-question\">A baseball of mass 0.145 kg leaves the bat at 38 m\/s. What is its kinetic energy?<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Kinetic energy, KE = \u00bd\u00b7m\u00b7v<sup>2<\/sup>.<\/p>\n<p>Step 2: KE = \u00bd \u00d7 0.145 kg \u00d7 (38 m\/s)<sup>2<\/sup> = \u00bd \u00d7 0.145 \u00d7 1,444 kg\u00b7m<sup>2<\/sup>\/s<sup>2<\/sup><\/p>\n<p>Step 3: KE = 104.69 J<\/p>\n<p><strong>Answer: KE = 105 J (3 s.f.)<\/strong><\/p>\n<\/div><\/details><\/div>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 3<\/div><div class=\"pf-problem-question\">A 65 kg cyclist travelling at 8.0 m\/s brakes to a stop in 4.0 s. Find the change in momentum and the average braking force.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Initial momentum, p = m\u00b7v = 65 kg \u00d7 8.0 m\/s = 520 kg\u00b7m\/s. Final momentum is zero.<\/p>\n<p>Step 2: Impulse equals change in momentum, so J = \u0394p = \u2212520 N\u00b7s (the minus sign means it opposes the motion).<\/p>\n<p>Step 3: From J = F\u00b7\u0394t, F = 520 N\u00b7s \/ 4.0 s = 130 N<\/p>\n<p><strong>Answer: \u0394p = 520 kg\u00b7m\/s in magnitude, average braking force = 130 N<\/strong><\/p>\n<\/div><\/details><\/div>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 4<\/div><div class=\"pf-problem-question\">A 2.0 kg block is released from rest and slides 4.0 m down a frictionless slope inclined at 30\u00b0. Find its speed at the bottom using both the force route and the energy route.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1 (force route): The acceleration along the slope is a = g\u00b7sin\u03b8 = 9.81 m\/s<sup>2<\/sup> \u00d7 sin30\u00b0 = 4.905 m\/s<sup>2<\/sup><\/p>\n<p>Step 2: Apply v<sup>2<\/sup> = u<sup>2<\/sup> + 2as with u = 0: v<sup>2<\/sup> = 0 + 2 \u00d7 4.905 m\/s<sup>2<\/sup> \u00d7 4.0 m = 39.24 m<sup>2<\/sup>\/s<sup>2<\/sup>, so v = 6.264 m\/s<\/p>\n<p>Step 3 (energy route): The vertical drop is h = 4.0 m \u00d7 sin30\u00b0 = 2.0 m. Setting m\u00b7g\u00b7h = \u00bd\u00b7m\u00b7v<sup>2<\/sup> and cancelling m gives v = sqrt(2\u00b7g\u00b7h) = sqrt(2 \u00d7 9.81 \u00d7 2.0) = 6.264 m\/s<\/p>\n<p><strong>Answer: v = 6.3 m\/s by both routes (2 s.f.)<\/strong><\/p>\n<\/div><\/details><\/div>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 5<\/div><div class=\"pf-problem-question\">A 40 kg crate is pulled 6.0 m across a floor by a 150 N force acting at 25\u00b0 above the horizontal. The coefficient of friction is 0.30. Find the work done by each force and the crate&#039;s final speed, starting from rest.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Normal force. The vertical component of the pull lifts part of the weight: N = m\u00b7g \u2212 F\u00b7sin\u03b8 = (40 \u00d7 9.81) \u2212 (150 \u00d7 sin25\u00b0) = 392.4 \u2212 63.39 = 329.01 N<\/p>\n<p>Step 2: Friction force, F_f = \u03bc\u00b7N = 0.30 \u00d7 329.01 N = 98.70 N<\/p>\n<p>Step 3: Work by the pull, W = F\u00b7d\u00b7cos\u03b8 = 150 N \u00d7 6.0 m \u00d7 cos25\u00b0 = 815.68 J. Work by friction = \u221298.70 N \u00d7 6.0 m = \u2212592.21 J<\/p>\n<p>Step 4: Net work = 815.68 \u2212 592.21 = 223.46 J. Setting this equal to \u00bd\u00b7m\u00b7v<sup>2<\/sup> gives v = sqrt(2 \u00d7 223.46 \/ 40) = 3.343 m\/s<\/p>\n<p><strong>Answer: 816 J by the pull, \u2212592 J by friction, final speed 3.3 m\/s<\/strong><\/p>\n<\/div><\/details><\/div>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 6<\/div><div class=\"pf-problem-question\">A 0.20 kg ball on a 0.75 m string is whirled in a horizontal circle at 3.0 revolutions per second. Find the angular velocity, the tangential speed and the centripetal force.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Angular velocity, \u03c9 = 2\u03c0\u00b7f = 2\u03c0 \u00d7 3.0 s<sup>\u22121<\/sup> = 18.85 rad\/s<\/p>\n<p>Step 2: Tangential speed, v = \u03c9\u00b7r = 18.85 rad\/s \u00d7 0.75 m = 14.14 m\/s<\/p>\n<p>Step 3: Centripetal force, F_c = m\u00b7v<sup>2<\/sup> \/ r = 0.20 kg \u00d7 (14.14 m\/s)<sup>2<\/sup> \/ 0.75 m = 53.30 N. Checking with F_c = m\u00b7\u03c9<sup>2<\/sup>\u00b7r gives 0.20 \u00d7 18.85<sup>2<\/sup> \u00d7 0.75 = 53.30 N, the same value.<\/p>\n<p><strong>Answer: \u03c9 = 18.8 rad\/s, v = 14.1 m\/s, F_c = 53.3 N<\/strong><\/p>\n<\/div><\/details><\/div>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 7<\/div><div class=\"pf-problem-question\">A 12 g bullet travelling at 350 m\/s embeds itself in a 2.0 kg block resting on a frictionless surface against a spring of stiffness 800 N\/m. Find the speed of the block just after impact and the maximum compression of the spring.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: Momentum is conserved through the impact. Before: p = 0.012 kg \u00d7 350 m\/s = 4.2 kg\u00b7m\/s<\/p>\n<p>Step 2: After the bullet embeds, the combined mass is 2.012 kg, so v = 4.2 \/ 2.012 = 2.088 m\/s<\/p>\n<p>Step 3: Kinetic energy of the combined mass, KE = \u00bd \u00d7 2.012 kg \u00d7 (2.088 m\/s)<sup>2<\/sup> = 4.384 J<\/p>\n<p>Step 4: This energy is stored in the spring, so \u00bd\u00b7k\u00b7x<sup>2<\/sup> = 4.384 J, giving x = sqrt(2 \u00d7 4.384 \/ 800) = 0.1047 m<\/p>\n<p><strong>Answer: v = 2.09 m\/s, maximum compression = 10.5 cm<\/strong><\/p>\n<p>Note the size of the energy loss: the bullet arrived with 735 J and only 4.4 J survived. Over 99 per cent went into heat and deformation \u2014 which is exactly why you cannot use energy conservation across an embedding collision.<\/p>\n<\/div><\/details><\/div>\n \n<div class=\"pf-problem\"><div class=\"pf-problem-num\">Problem 8<\/div><div class=\"pf-problem-question\">A solid disc flywheel of mass 8.0 kg and radius 0.30 m is spun from rest to 240 revolutions per minute in 5.0 s by a constant torque. Find its moment of inertia, angular acceleration, the applied torque and its rotational kinetic energy.<\/div><details><summary>Show Solution<\/summary><div class=\"pf-problem-solution\">\n<p><strong>Solution:<\/strong><\/p>\n<p>Step 1: For a solid disc, I = \u00bd\u00b7m\u00b7r<sup>2<\/sup> = \u00bd \u00d7 8.0 kg \u00d7 (0.30 m)<sup>2<\/sup> = 0.36 kg\u00b7m<sup>2<\/sup><\/p>\n<p>Step 2: Convert the final speed: \u03c9 = 240 \u00d7 2\u03c0 \/ 60 = 25.13 rad\/s<\/p>\n<p>Step 3: Angular acceleration, \u03b1 = \u03c9 \/ t = 25.13 \/ 5.0 = 5.027 rad\/s<sup>2<\/sup><\/p>\n<p>Step 4: Torque, \u03c4 = I\u00b7\u03b1 = 0.36 kg\u00b7m<sup>2<\/sup> \u00d7 5.027 rad\/s<sup>2<\/sup> = 1.81 N\u00b7m. Rotational kinetic energy, KE = \u00bd\u00b7I\u00b7\u03c9<sup>2<\/sup> = \u00bd \u00d7 0.36 \u00d7 25.13<sup>2<\/sup> = 113.7 J<\/p>\n<p><strong>Answer: I = 0.36 kg\u00b7m<sup>2<\/sup>, \u03b1 = 5.03 rad\/s<sup>2<\/sup>, \u03c4 = 1.81 N\u00b7m, KE = 114 J<\/strong><\/p>\n<\/div><\/details><\/div>\n \n<h2>Frequently Asked Questions<\/h2>\n \n<details class=\"pf-faq-item\"><summary>What are the most important mechanics formulas to memorise?<\/summary><div class=\"pf-faq-item-answer\">\n \nSix carry most of the load: F = ma, the four constant-acceleration equations, KE = \u00bdmv<sup>2<\/sup>, PE = mgh, W = Fd\u00b7cos\u03b8 and p = mv. Learn these properly, including what every symbol means and its SI unit, and the rest of the sheet can be looked up. Memorising equations you cannot yet select between adds no marks.\n \n<\/div><\/details>\n \n<details class=\"pf-faq-item\"><summary>How do you identify the variables in a mechanics problem?<\/summary><div class=\"pf-faq-item-answer\">\n \nRead the question once and write each number down with its symbol and SI unit before doing anything else. Convert grams to kilograms and minutes to seconds as you go. Then write the symbol of the quantity being asked for. That short list of knowns and one unknown is exactly what you match against the formula tables.\n \n<\/div><\/details>\n \n<details class=\"pf-faq-item\"><summary>How do I know whether to use energy or momentum?<\/summary><div class=\"pf-faq-item-answer\">\n \nUse momentum when two objects interact and you know the masses and velocities involved, especially in a collision. Use energy when you are given heights, speeds or distances and no time. Momentum is conserved in every collision; kinetic energy is only conserved in elastic ones, so momentum is the safer choice whenever objects stick together.\n \n<\/div><\/details>\n \n<details class=\"pf-faq-item\"><summary>How do you check a mechanics formula using units?<\/summary><div class=\"pf-faq-item-answer\">\n \nReplace every symbol with its SI unit and simplify. For kinetic energy, \u00bdmv<sup>2<\/sup> becomes kg \u00d7 (m\/s)<sup>2<\/sup> = kg\u00b7m<sup>2<\/sup>\/s<sup>2<\/sup>, which is the joule, so the formula is dimensionally sound. If the units do not match the quantity you wanted, the formula is wrong. Note that this catches wrong formulas but not wrong numerical factors.\n \n<\/div><\/details>\n \n<details class=\"pf-faq-item\"><summary>Do mechanics formulas only work in SI units?<\/summary><div class=\"pf-faq-item-answer\">\n \nThe equations themselves work in any consistent set of units, but every constant on this page assumes SI. Mixing units without converting is the most common source of wrong answers in mechanics. Convert everything to kilograms, metres and seconds before substituting, and your result will come out in newtons, joules or watts automatically.\n \n<\/div><\/details>\n \n<details class=\"pf-faq-item\"><summary>When does F = ma stop working?<\/summary><div class=\"pf-faq-item-answer\">\n \nF = ma holds to excellent accuracy for everyday speeds and everyday sizes. It breaks down at speeds approaching the speed of light, where relativistic mechanics takes over, and at atomic scales, where quantum mechanics applies. For anything you can see, throw or drive, the classical mechanics formulas on this page are correct.\n \n<\/div><\/details>\n \n","protected":false},"excerpt":{"rendered":"<p>Open any mechanics textbook to the back cover and you will find the same thing: a wall of equations, printed small, with no clue about which one to reach for. Most students copy it out, memorise it, and still stall on the first line of a problem. That stall is almost never a memory failure. &#8230; <a title=\"Mechanics Formulas Cheat Sheet\" class=\"read-more\" href=\"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/mechanics-formulas\/\" aria-label=\"Read more about Mechanics Formulas Cheat Sheet\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":756,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-755","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mechanics"],"_links":{"self":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/755","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/comments?post=755"}],"version-history":[{"count":6,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/755\/revisions"}],"predecessor-version":[{"id":1490,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/posts\/755\/revisions\/1490"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media\/756"}],"wp:attachment":[{"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/media?parent=755"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/categories?post=755"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicsfundamentalsinfo.com\/blog\/wp-json\/wp\/v2\/tags?post=755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}