Collisions in a straight line: signs are everything

5 minAS Level3.3.2

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can solve one-dimensional collision problems with conservation of momentum, using a sign convention for direction, whether the objects stick together or bounce apart.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$m_1$	mass of the first object
$u_1$	initial velocity of the first object
$m_2$	mass of the second object
$u_2$	initial velocity of the second object
$v_1$	final velocity of the first object
$v_2$	final velocity of the second object
$u_A$	initial velocity of object A
$u_B$	initial velocity of object B
$v_A$	final velocity of object A

In a straight line, a vector is a number with a sign

Two identical cars drive at each other at the same speed. What is their total momentum?

Zero. One momentum is $+mv$ , the other $-mv$ , and they cancel.

That's the whole trick of 1D momentum problems: along a line, a vector's direction is carried by its sign. Forget the sign and you get answers that are confidently, spectacularly wrong.

The four-step recipe

Sketch before and after, with an arrow for each velocity.
Choose a positive direction (say "right is positive") and write it down.
Write total momentum before = total momentum after, with signs: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ ( $u$ = velocities before, $v$ = velocities after). If they stick together, the right-hand side is $(m_1 + m_2)v$ .
Solve and interpret. A negative answer isn't a mistake: it means the velocity is in the negative direction. Say the direction in words.

Worked example: Head-on, locked together

A car of mass $1500\ \text{kg}$ travelling east at $20\ \text{m s}^{-1}$ collides head-on with a van of mass $2500\ \text{kg}$ travelling west at $10\ \text{m s}^{-1}$ . They lock together. Find their velocity just after the collision.

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

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