Intuition The one core ideaA collision has two unbreakable bookkeeping rules — momentum (a directional arrow-total that never changes) and kinetic energy (a single number of "motion-cash" that, for an elastic hit, also never changes). When you write both rules down for two equal balls and combine them, geometry forces the two outgoing paths into a perfect right angle.
Before you can derive the 90° result on the parent page, every arrow, symbol, and squiggle there must mean something to you. This page builds each one from nothing. Read top to bottom — each block uses only what came before it.
A vector is an arrow: it has a length (how much) and a direction (which way). We write it with a little arrow on top, like u 1 ∣ u 1 ∣ u 1
Look at the figure below. The lavender arrow is a velocity. It leans, so it has direction. It is long, so the speed is large. Two facts, one arrow.
Intuition Why the topic needs vectorsA ball doesn't just have a speed — it moves somewhere . "5 m/s" is useless until you say "5 m/s to the right." Collisions in 2D are all about directions splitting apart, so every velocity must be an arrow, not a number.
Any arrow can be described by its shadow along two perpendicular directions : how far right (x y components . We write them as a pair, a = ( a x , a y ) a x a y
In the figure the coral arrow v 1 θ 1
the flat side (adjacent to θ 1 x-component v 1 cos θ 1
the upright side (opposite θ 1 y-component v 1 sin θ 1
sin cos θ cos θ = hypotenuse adjacent side , sin θ = hypotenuse opposite side
Cosine = "how much of the arrow points along the base." At θ = 0 ∘ cos 0 ∘ = 1 Sine = "how much points upward ." At θ = 9 0 ∘ sin 9 0 ∘ = 1
Intuition Why we need componentsYou can't add two slanted arrows by eye. But you can add all the right-parts together and all the up-parts together separately. Splitting into components turns one hard vector equation into two easy number equations — exactly the "x and y" split on the parent page.
Common mistake "Sine is always the vertical one."Why it feels right: In the picture above, sine gave the height. The fix: sin cos which angle you measure from . Opposite/adjacent are relative to that angle. Always draw the triangle and label the angle first.
The formulas v cos θ v sin θ every direction, because sin cos change sign depending on where the arrow points.
Intuition Why this matters hereOn the parent page the struck ball flies below the line — its y-component is negative . That is exactly why the parent's y-momentum equation has a minus sign: v 2 sin θ 2
Conservation of momentum literally adds velocity arrows, so we must nail down what "+
Definition Vector additionTo add two arrows a + b b a a b just add the parts :
a + b = ( a x + b x , a y + b y )
Intuition Why addition, and why component-wise"Total momentum" means combine the arrows of both balls into one. Geometrically that's tip-to-tail. But adding slanted arrows by eye is hopeless, so we add the right-parts together and the up-parts together — that is the deep reason components exist. The parent's u 1 = v 1 + v 2
Common mistake "Add the lengths:∣ a + b ∣ = ∣ a ∣ + ∣ b ∣ Why it feels right: Numbers add that way. The fix: Only true if the arrows point the same way. Point them apart and the sum is shorter than the two lengths added — that shortening is precisely what makes the collision triangle a real triangle.
Why is this true? The two components are the legs of a right triangle whose hypotenuse is the arrow of length v 1 ( v 1 cos θ 1 ) 2 + ( v 1 sin θ 1 ) 2 = v 1 2 . v 1 2 v 1 2 + v 2 2 = u 1 2 is this identity.
The parent page uses several velocity letters. Fix their meanings now so no subscript surprises you.
Definition Velocity roster
u 1 before the collision (the incoming ball); its length is u 1 u 2 before . On the parent page ball 2 is at rest , so u 2 = 0 u 2 = 0 v 1 after ; its length (speed) is v 1 v 2 after ; its length is v 2
Rule of thumb: u v v 1 2 + v 2 2 = u 1 2
Momentum of one particle is its mass times its velocity arrow:
p = m v vector (mass is just a number, so m v v
Intuition Why momentum, and why it's conservedDuring a collision the two balls push on each other, but nothing outside pushes on the pair. Newton's third law says their mutual pushes are equal-and-opposite, so they cancel in the total. The total momentum arrow of the pair therefore cannot change — this is Conservation of Momentum . That is the single most important rule the parent page leans on.
Because momentum is a vector, "total momentum unchanged" is really two promises (using the addition rule from §3): the total right-part is unchanged and the total up-part is unchanged. That is where the parent's two equations (x and y) come from.
Definition Kinetic energyKinetic energy is the energy stored in motion:
K E = 2 1 m v 2 scalar — a plain number, no direction (note it uses v 2 length of the velocity, squared). See Kinetic Energy .
Definition Elastic vs inelastic
Elastic collision: the total kinetic-energy number is the same after as before. Nothing lost to heat, sound, or dents.Inelastic collision: some KE is lost. See Inelastic collisions .
Intuition Why energy gives only ONE equationMomentum is an arrow, so it splits into x and y — two equations. Energy is just a number, so it gives one equation. In 2D that's 2 + 1 = 3
Common mistake "Momentum and energy are the same thing."Why it feels right: Both grow when the ball speeds up. The fix: Momentum is m v 2 1 m v 2
This is the star tool of the parent proof, so we build it carefully.
Definition Dot product — two equal definitionsThe dot product of two arrows a b geometric form and an algebraic form, and they always agree:
a ⋅ b = geometric ∣ a ∣ ∣ b ∣ cos ϕ = algebraic (component) a x b x + a y b y ϕ between the two arrows, and ( a x , a y ) ( b x , b y ) Dot Product .
Intuition Why the dot product, and not some other tool?We want a machine that eats two arrows and tells us the angle between them — precisely the question "are the outgoing balls perpendicular?" The dot product is that machine. Its magic value is when the answer is 9 0 ∘ cos 9 0 ∘ = 0 ⟹ a ⋅ b = 0. dot product = 0 " is the same statement as "the arrows are at a right angle ." When the parent page reaches v 1 ⋅ v 2 = 0 proven the 90°.
a ⋅ b Why it feels right: You started with two arrows, so the answer should be an arrow too. The fix: The dot product spits out a plain number (a scalar). That is exactly why it's useful for energy-like bookkeeping and for testing angles.
Definition Handy values (note: some are rounded)
θ sin θ cos θ
0 ∘ 0 1
3 0 ∘ 0.5 0.866 … ≈ 0.87
3 7 ∘ ≈ 0.6 ≈ 0.8
5 3 ∘ ≈ 0.8 ≈ 0.6
9 0 ∘ 1 0
The 3 7 ∘ /5 3 ∘ sin 3 7 ∘ = 0.6018 … 0.6 0.8 approximations chosen because they make the tidy 3 4 5 0.3% 9 0 ∘
The map below is a dependency chart : read an arrow "A → B A B vector ), follow the arrows down, and every path eventually funnels into the 90 degree angle result at the bottom. If the diagram doesn't render on your device, the same order is: vector → components → (sign/quadrant rules, addition, sin/cos identity); vector → momentum → conservation → two equations; kinetic energy → elastic → one energy equation; vector → dot product → "dot = 0 means 90°". All of those meet at the final result.
Kinetic energy half m v squared
Elastic means KE conserved
Dot product measures angle
Dot equals zero means 90 degrees
Test yourself — you are ready for the parent page when you can answer each:
What two facts does a single vector arrow carry? Its length (magnitude) and its direction.
How do you write a vector in components? As a pair
a = ( a x , a y ) : its x-part and y-part.
How do you split an arrow into components? Drop perpendicular shadows onto the x and y axes, forming a right triangle; sides are v cos θ v sin θ
When are the components negative? When the arrow points left (cos θ < 0 sin θ < 0
How do you add two vectors, in a picture and in components? Tip-to-tail (start of the first to tip of the second); in components add part-by-part, ( a x + b x , a y + b y )
Is ∣ a + b ∣ ∣ a ∣ + ∣ b ∣ No — only if they point the same way; otherwise the sum is shorter.
What does cos θ Adjacent side divided by hypotenuse — how much of the arrow points along the base.
State the Pythagoras identity for sine and cosine. sin 2 θ + cos 2 θ = 1
What do u 1 , u 2 , v 1 , v 2 u v u 2 = 0
Write the momentum of one particle. Why is total momentum conserved in a collision? The balls' mutual pushes are equal and opposite (Newton's third law) and cancel; no outside force acts.
Why does momentum give two equations in 2D but energy only one? Momentum is a vector (x and y parts); kinetic energy is a scalar (one number).
Write kinetic energy and say what "elastic" means. K E = 2 1 m v 2
Give BOTH definitions of the dot product. Geometric
a ⋅ b = ∣ a ∣∣ b ∣ cos ϕ and algebraic
a ⋅ b = a x b x + a y b y .
State the distributive property that lets you square a sum. ( v 1 + v 2 ) ⋅ ( v 1 + v 2 ) = v 1 2 + 2 v 1 ⋅ v 2 + v 2 2 .
What does a ⋅ b = 0 They are perpendicular — at 9 0 ∘ cos 9 0 ∘ = 0
What does a ⋅ a ∣ a ∣ 2 , the length squared — the "square both sides" trick.
