Intuition The ONE core ideaSome forces are honest — they give back every joule you spend against them (gravity). Others are thieves — friction and air drag always steal energy and turn it into heat, and how much they steal depends on the whole path you took, not just where you started and ended. This page builds every symbol, arrow, and word you need before that idea can even be stated.
Before you can read the parent note , you must own its alphabet. We build each piece from nothing, in an order where every symbol is earned before it is used.
Two of these appear inside almost every formula on this page, so we define them first , before anything is built on them.
Definition Mass, gravity, height
m mass : how much "stuff" an object contains, measured in kilograms (kg). A brick has more m g gravitational acceleration : how fast Earth speeds up a falling object, g ≈ 9.8 m/s 2 h height : how high an object sits above some chosen floor level, measured in metres (m).
Intuition Why define these nowThe weight of an object — how hard gravity pulls it down — is the product m g h m g h m g m g h
v F r vector is an arrow: it has a length (how much) and a direction (which way). We draw it as a letter with a tiny arrow on top, like v how fast AND toward where .
Look at Figure s01 . The red arrow is a velocity vector v
Figure s01 — a velocity vector: length encodes speed, tilt encodes direction.
Intuition Why the topic needs vectorsFriction and drag are defined by the sentence "the force points opposite to the velocity ." That sentence is about directions — you literally cannot say it without arrows. So vectors are the first brick.
r position vector: an arrow from the origin to where the object is right now.d r tiny step along the object's path — an arrow so short it's almost a dot, pointing the way the object is moving at that instant.
v ^ unit vector is an arrow of length exactly 1 direction . We write it with a hat : v ^ v
So v = v v ^ v speed .
Intuition Why we need the hatThe parent writes drag as a force pointing backward along the motion. To say that in symbols we need a piece that carries only the direction of the velocity — that is exactly the hat v ^
Here is the single most important tool on this page. When a force F d r helping the motion ?
Figure s02 — three panels: force aligned (0 ∘ 9 0 ∘ 18 0 ∘
The dot product of two arrows multiplies their lengths and then keeps only the part that points the same way :
F ⋅ d r = ∣ F ∣ ∣ d r ∣ cos ϕ ϕ between the two arrows, and cos ϕ
Intuition Why the dot product and not plain multiplication?Plain multiplication ignores direction. But a sideways push does no work — only the part of the force along the motion counts. The dot product is the exact tool that answers "how much of this force is doing useful pushing?" — that is precisely the question "work" asks.
The three cases you must never forget (the three panels of Figure s02 , left to right):
Same direction (ϕ = 0 cos 0 = 1 Right angle (ϕ = 9 0 ∘ cos 9 0 ∘ = 0 Opposite direction (ϕ = 18 0 ∘ cos 18 0 ∘ = − 1 negative . Force fights the motion.
∫ ∫ add up a quantity over a whole path, one tiny piece at a time." ∫ F ⋅ d r d r
Think of it as a total. Each step contributes a sliver of work; the integral is the whole pile.
Definition Closed-loop integral∮ ∮ returns to where it started — a closed loop. ∮ F ⋅ d r
Figure s03 — a closed loop: start equals end, yet friction (red arrows) opposes every leg.
Intuition Why the loop symbol is the whole testThe parent's definition of conservative vs non-conservative is a single equation: ∮ F ⋅ d r = 0 = 0 every leg of the loop, so its slivers are all negative and the total can never cancel to zero. You cannot even read that definition without ∮
L total path length : the full distance travelled, the whole squiggly line's length — not the straight-line displacement.displacement = the single straight arrow from start to finish. On a closed loop displacement is zero, but L
Definition The energy quantities
W work : energy transferred by a force, measured by the dot-product-integral above. Units: joules (J).K E = 2 1 m v 2 kinetic energy (in joules): energy of motion . Half the mass m v U potential energy (in joules): stored energy of position — e.g. gravity's U = m g h conservative forces have a U E m ec h = K E + U mechanical energy (in joules): motion-energy plus stored-energy, the running total that friction slowly drains.W n c non-conservative work : the work done specifically by the thief forces (friction, drag) — always negative, since they always oppose motion. Units: joules (J).Q heat (in joules): the energy that friction/drag steal and dump as warmth. The "lost" mechanical energy reappears here — it never truly vanishes.
The triangle Δ "change in" : Δ K E = K E final − K E initial
f k = μ k N
N normal force (in newtons): the surface pushing straight out (perpendicular) on the object, holding it up. On flat ground N = m g N = m g cos θ μ k coefficient of kinetic friction : a plain number (no units) saying how "grippy" the sliding surfaces are. Ice ≈ 0.05 0.8 f k = μ k N size of kinetic friction (in newtons): grippiness times how hard surfaces press together. Its vector form is f k = − f k v ^ θ incline angle : the tilt of a ramp, measured from horizontal.
Figure s04 — a block on an incline: weight m g m g cos θ m g sin θ
N N
Definition Symbols in the drag formulas
v speed (the plain number). Drag depends on it — faster means more push-back.b drag constant for the slow (linear) model F d = b v C d drag coefficient : a shape number (streamlined ≈ small, flat plate ≈ large), no units.ρ air density (Greek "rho"): how much air-mass sits in each cubic metre (kg/m 3 A frontal area (in m 2 v t terminal velocity : the steady speed reached when drag exactly balances gravity, so acceleration stops.
Intuition Why two drag modelsSlow through a fluid: you feel its stickiness → force grows like v slam air masses aside, and both how-much-air and how-hard scale with v v 2
Given numbers mass m gravity g height h
Work energy quantities KE U Q
Vectors arrow with length and direction
Unit vector the hat direction only
Dot product force times aligned motion
Integral add up work over a path
Closed loop integral the loop test
Path length L versus displacement
Mechanical energy bookkeeping Wnc
Normal force and friction number
Speed density area drag constants
Drag force models and terminal velocity
Non-conservative forces friction and air drag
Every foundation above flows into the topic box. If any incoming arrow is unclear to you, revisit that section before opening the parent note.
Cover the right side and test yourself. If you can answer all, you are ready.
What do m g h m g ≈ 9.8 m/s 2 h m g m g h
What does a hat like v ^ v The hat = direction only (length 1); the plain letter = the speed number. Together
v = v v ^ .
What does the dot product F ⋅ d r How much of the force points
along the motion — the work done on that step; it equals
∣ F ∣∣ d r ∣ cos ϕ .
What is cos ϕ cos 18 0 ∘ = − 1
What does the stretched-S ∫ ∮ ∫ ∮
Difference between path length L L
Write K E Δ Δ K E K E = 2 1 m v 2 Δ
What is W n c Δ E m ec h Q W n c W n c = Δ E m ec h Q = − W n c
Derive the normal force N θ Split m g m g cos θ N m g sin θ N = m g cos θ θ = 0 N = m g
What does μ k The coefficient of kinetic friction — a grippiness number, dimensionless (no units).
Write the high-speed drag force, strength and vector form. Strength
F d = 2 1 C d ρ A v 2 ; vector
F d = − 2 1 C d ρ A v 2 v ^ (opposes motion). Low speed:
F d = b v .
Write terminal velocity v t Low speed:
v t = m g / b . High speed:
v t = 2 m g / ( C d ρ A ) (set drag = weight).
Where does the "lost" mechanical energy go, and by what symbol? Into heat Q Q = − W n c
Parent topic — Non-conservative forces (Hinglish) — the note this page prepares you for.Conservative forces & potential energy — where U Work–Energy theorem — where W = Δ K E Friction — static & kinetic — deeper on μ k N Terminal velocity & projectile with drag — uses every drag symbol here. 
