The key formulas of mechanics on one physics sheet, grouped by the topics of AP Physics 1 and A level mechanics: kinematics, forces, energy, momentum, rotation, oscillations, gravitation and fluids.
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The key formulas of a mechanics course on one page, grouped by the topics of AP Physics 1 and of A level mechanics. Symbols and conditions are named under each formula, and the sheet ends with the constants. Formulas with a calculator link to it, so you can try your own values. Switch to quiz mode to hide the formulas and test yourself, or print the whole sheet.
For a constant acceleration only.
s displacementu initial velocityv final velocitya accelerationt time
d distanceΔx displacementt, Δt time
ā average accelerationΔv change in velocityt time
No air resistance. T, H and R only when the projectile lands at the height it was launched from.
u launch speedθ launch anglex, y horizontal and vertical distance from the launch pointt timeg acceleration due to gravity
ΣF net forcem massa accelerationg acceleration due to gravity
μ coefficient of frictionN normal force
W weightN normal forceθ angle of the slope
k spring constantx extension
v speedr radiusT periodω angular velocity
m massv speed
h heightk spring constantx extension
F forced displacementθ angle between force and motion
W workΔt timeθ angle between force and motion
Mechanical energy, when only gravity does work: no friction, drag or other force adding or removing energy.
p momentumJ impulseF̄ average force
u velocity beforev velocity after
m massx position
The last two: for a constant angular acceleration only.
ω angular velocityα angular accelerationΔθ angle in radians
r distance from the pivotI moment of inertiaM massd distance between the two parallel axes
I moment of inertiaω angular velocityL angular momentum
m massk spring constant
For small swings only, up to about 15°.
L lengthg acceleration due to gravity
A amplitudeT period
G gravitational constantr distance between the centersM mass of the planet
For a circular orbit.
M mass of the planetr orbit radius
ρ densitym massV volume
A areaρ density of the fluidh depthV volume under the surface
Steady flow of a fluid that does not compress, with no friction.
A areav flow speedy height
The five kinematic equations link five quantities: the displacement s, the initial velocity u, the final velocity v, the acceleration a and the time t. That is why UK courses call them the SUVAT equations. Each equation leaves out exactly one of the five.
So write down the three values you know and the one you want. The letter left over is the one you neither have nor need, and the equation without it is the one to use. A car braking from 20 m/s to a stop in 50 m, with no time given and none asked for, calls for v² = u² + 2as, the equation without t: 0 = 20² + 2a · 50, so the car slows at 4 m/s².
The equations hold only while the acceleration is constant. A falling stone qualifies (until air resistance matters); a car that speeds up and then brakes has to be split into two parts, each with its own acceleration.
Displacement, velocity, acceleration, force and momentum all have a direction, and the sheet writes them as plain numbers. Pick one direction as positive at the start of a problem and keep it. If up is positive, a ball thrown upward has a positive u and an acceleration of −9.81 m/s², and its displacement turns negative once it falls below the point it was thrown from.
The minus sign in Hooke's law, F = −kx, says the same thing: the spring pushes back against the way it was stretched. When you only want the size of the force, use kx.
Physics courses agree on the formulas and disagree on the letters. This sheet uses the letters of the kinematics calculator and of most UK courses. AP Physics 1 writes the same ideas this way:
Every formula on the sheet works in SI units: metres, kilograms and seconds. The others are built from those three. A newton is the force that gives 1 kg an acceleration of 1 m/s², a joule is the work of 1 N over 1 m, and a watt is 1 J every second. Convert kilometres per hour, grams and minutes before you substitute, and the answer comes out in the right unit with no further work.
The physics calculators solve these formulas for any value you leave blank and show each step, from the force calculator to the momentum calculator. For circuits, fields and waves, the electricity formula sheet is the next page.
Not the official one: the College Board gives you its own sheet in the exam, so practise with that. This is a free supplement that covers the same topics (kinematics, forces, work and energy, momentum, rotation, oscillations and fluids) and writes the conditions beside each formula. It writes u for the initial velocity where the AP sheet writes v₀; the section on notation lists the main differences.
v = u + at, s = ½(u + v)t, s = ut + ½at², v² = u² + 2as and s = vt − ½at². They hold only when the acceleration is constant, and each one leaves out one of the five quantities s, u, v, a and t, which is how you pick the one to use.
Mass is how much matter an object has, measured in kilograms, and it is the same everywhere. Weight is the force gravity pulls it with, W = mg, measured in newtons. A 70 kg person weighs about 687 N on Earth and about 113 N on the Moon, with the same 70 kg of mass.
Momentum is conserved in every collision with no outside force, but kinetic energy is conserved only in an elastic one. When objects dent, heat up or stick together, some kinetic energy turns into other forms. A collision where the objects stick together loses the most.
A heavier bob needs more force to speed up, but gravity pulls it harder by exactly the same factor, so the two cancel and T = 2π√(L/g) depends only on the length and on g. The formula is an approximation that holds for small swings, up to about 15°.
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