Elastic collisions — 1D - solve for final velocities: Foundations

Before you can use the parent note (topic here), you must be fluent in every symbol it throws at you. This page builds each one from absolute zero, in the order they depend on each other. Nothing is assumed.

0. The stage: a straight line and a direction

Everything happens on a frictionless straight track. Because it is a line, motion has only two possible directions: one way or the other.

Why the topic needs it: the whole derivation is algebra with $+$ and $-$ numbers. If you drop a sign, "moving left" secretly becomes "moving right" and every later equation lies to you. The number line is the physics.

1. Mass — the symbols $m_{1}$ and $m_{2}$

The little numbers below — the subscripts — are just name tags:

$m_{1}$ ::: reads "m-one", the mass of object 1.
$m_{2}$ ::: reads "m-two", the mass of object 2.

They are not multiplication and not powers. They only say which object we mean.

Why the topic needs it: the outcome of a crash depends entirely on the ratio of the two masses. A heavy cart hitting a light one behaves totally differently from two equal carts. The subscripts let us keep the two objects straight through the algebra.

2. Velocity — $u$ (before) and $v$ (after)

We use two different letters so we never confuse "before" and "after":

Formula The four velocity symbols

Symbol	Reads	Means
$u_{1}$	u-one	object 1's velocity before
$u_{2}$	u-two	object 2's velocity before
$v_{1}$	v-one	object 1's velocity after
$v_{2}$	v-two	object 2's velocity after

Memory hook: $u$ comes before $v$ in the alphabet, just like before comes before after in time.

Why the topic needs it: the entire problem is "given $u_{1}, u_{2}$ , find $v_{1}, v_{2}$ ." These four symbols are the question and the answer.

3. Momentum — the symbol $p = m v$

Now we combine mass and velocity into one quantity.

Why the topic needs it — and why this tool, not just "speed": we want a quantity that is conserved (stays constant) in a collision. Plain speed is not conserved. Momentum is — because the pushes the two carts give each other are equal and opposite (Newton's third law), so what one gains the other loses, and the total never changes. See Conservation of Momentum for the full "why". This gives us Equation 1 of the two we need: $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$ Read it as: (total momentum before) = (total momentum after).

4. Kinetic energy — the symbol $\frac{1}{2} m v^{2}$

We need a second rule, and it must be different from momentum (two identical rules would just repeat, giving us nothing new).

Two features to notice, because they change how the algebra behaves:

Why the topic needs it — why this second tool: an "elastic" collision is defined as one where no kinetic energy turns into heat, sound, or dents (see Inelastic Collisions — 1D for the case where it does). So total KE before = total KE after. This is Equation 2: $\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}$

Recall Why two equations = solvable

Two unknowns ( $v_{1}, v_{2}$ ) need two independent facts. Momentum gives one; kinetic energy gives a genuinely different one (it has squares). Two equations, two unknowns ⇒ exactly one answer. Two unknowns need how many independent equations? ::: Two.

5. Notation you must read fluently

The parent note uses a few shorthands. Decode them once, here.

6. Relative velocity — the approach and separation speeds

Why the topic needs it: the punchline of the whole derivation is that for elastic 1D collisions, approach speed = separation speed ( $u_{1} - u_{2} = v_{2} - v_{1}$ ). This linear rule replaces the ugly squared energy equation and makes solving easy. It is also the $e = 1$ case of the Coefficient of Restitution.

7. How it all fits together

Every box on the left is a foundation from this page; the arrows show how they feed the final answer the parent note computes.

Equipment checklist

Test yourself — cover the right side and answer out loud.

If rightward is positive, how do you write "moving left at 2 m/s"?

- 2

(velocities are signed).

What does the subscript in

m_{2}

mean?

A name tag — "the mass of object 2", not multiplication or a power.

Difference between

u

and

v

in this topic?

u

= velocity before the collision,

v

= velocity after.

Define momentum and give its formula.

"How much motion";

p = m v

(signed).

Why is momentum conserved in a collision?

The two objects push each other equally and oppositely, so the total never changes.

Define kinetic energy and give its formula.

Energy of motion;

\frac{1}{2} m v^{2}

Why does kinetic energy ignore direction?

It uses

v^{2}

, and squaring kills the sign:

(- v)^{2} = v^{2}

What makes a collision "elastic"?

No kinetic energy is lost (no heat/sound/dents); total KE is conserved.

Factor

a^{2} - b^{2}

(a - b) (a + b)

— the difference of squares.

What is the "approach speed"?

u_{1} - u_{2}

, the relative velocity — how fast the objects close in on each other.

Why do we need TWO conservation laws?

There are two unknowns (

v_{1}, v_{2}

); two independent equations pin them down.