Linear momentum

Linear momentum p = mv as a vector quantity in one dimension — how the sign carries the direction of motion, how to combine the momenta of several particles, and how to find a change in momentum by working with velocities (with signs) rather than speeds.

Mechanics · Momentum

Linear momentum

Why it works

The linear momentum of a particle is the product of its mass and its velocity:

p = mv.

Mass is measured in kilograms and velocity in metres per second, so momentum has units of

\text{kg}\,\text{m}\,\text{s}^{-1}

(equivalently

\text{N}\,\text{s}

). Because velocity is a vector, momentum is a vector too — in one dimension that means the sign of

p

records the direction of motion.

Choose a positive direction first. In this course all momentum problems are in a straight line, so before writing anything down, decide which way counts as positive. A particle moving the other way then has a negative velocity, and therefore a negative momentum. This single habit — velocities carry signs, speeds do not — is what makes momentum calculations come out right.

Combining momenta. The total momentum of several particles moving along the same line is the sum of their momenta, each with its own sign:

p_{\text{total}} = m_1 v_1 + m_2 v_2 + \cdots

Two particles heading towards each other partly cancel; you subtract, you do not add their magnitudes.

Change in momentum. The change in a particle's momentum is

\Delta p = mv_{\text{final}} - mv_{\text{initial}} = m\,(v_{\text{final}} - v_{\text{initial}}),

and again the velocities carry signs. If a ball bounces back off a wall, its velocity reverses sign, so the change in momentum is much larger than a naïve "final speed minus initial speed" would suggest — that reversal is the whole point.

Momentum is not the same as kinetic energy: momentum is

mv

(linear in

v

, a signed vector), while kinetic energy is

\tfrac12 mv^2

(a positive scalar). Keep the two apart.