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Linear Momentum

Source lecture(s): SC133 Lec 10

Intuition

Which is harder to stop: a bicycle at 30 km/h or a truck at 30 km/h? Same velocity, very different "quantity of motion." Momentum \(\vec p = m\vec v\) is that quantity — mass times velocity, the measure of how much oomph a moving body carries and how much force-time it takes to change it. Newton wrote his second law in terms of it, and unlike velocity or kinetic energy, total momentum survives even the most violent collisions.

Definition and Newton's second law, properly

\[\vec p = m\vec v \qquad\qquad \boxed{\;\sum\vec F = \frac{d\vec p}{dt}\;}\]

For constant mass this reduces to \(\sum\vec F = m\vec a\); the momentum form is more general (rockets shed mass; relativity redefines \(p\) but keeps this law).

Conservation of momentum

For a system with zero net external force:

\[\vec P_\text{total} = \sum_i m_i \vec v_i = \text{constant}\]

The proof is Newton's third law: internal forces come in canceling pairs, so they can redistribute momentum among parts but never change the total. This is the workhorse of collision analysis — during the brief violence of impact, external forces (gravity, friction) are negligible next to the enormous internal ones, so momentum passes through the collision unchanged even when kinetic energy does not.

Deeper view (for later courses): momentum conservation follows from the homogeneity of space — physics doesn't care where the experiment happens.

Momentum vs kinetic energy

Momentum \(\vec p\) Kinetic energy \(K\)
Type vector scalar
Formula \(mv\) \(\tfrac12 mv^2 = p^2/2m\)
In collisions always conserved (isolated) conserved only if elastic
Can cancel between bodies? yes (opposite directions) never (always ≥ 0)

Two identical cars in a head-on crash: total momentum zero before and after; total kinetic energy large before, mangled metal after.

Worked example: recoil

A 4 kg rifle fires a 10 g bullet at 500 m/s. Recoil speed?

\[0 = m_b v_b + m_r v_r \Rightarrow v_r = -\frac{0.01 \times 500}{4} = -1.25\,\text{m/s}\]

Momenta are equal and opposite; kinetic energies are not — the bullet carries \(K_b/K_r = m_r/m_b = 400\) times more. (Same split powers rockets and explains why the lighter fragment always gets the energy.)

Common mistakes

  • Conserving momentum with external forces present. A ball bouncing off the floor does not conserve its momentum — Earth intervenes. Choose the system so external forces vanish (or act negligibly during the event).
  • Treating momentum as a scalar. Head-on momenta subtract; perpendicular momenta combine as vectors, component by component.
  • Assuming KE conservation because momentum is conserved. Independent claims — see collisions.

Knowledge graph position

Prerequisites: Newton's laws. Leads to: Collisions & impulse, [rocket propulsion], angular momentum (the rotational analogue).

Quiz

Q1 (computational). A 0.15 kg ball at 40 m/s is caught and stopped. Magnitude of the momentum change?

Answer

\(|\Delta p| = 0.15 \times 40 = 6\,\text{kg·m/s}\) — delivered to the catcher's hand over the stopping time (see impulse).

Q2 (conceptual). Can a system have zero total momentum but large kinetic energy?

Answer

Yes — two equal masses moving oppositely, a spinning object, a hot gas. Momenta cancel as vectors; kinetic energies add as positive scalars.

Q3 (multiple choice). A heavy truck and light car have equal kinetic energy. Which has more momentum? (a) truck (b) car (c) equal

Answer

(a). \(p = \sqrt{2mK}\): at fixed \(K\), momentum grows with mass — the mirror image of the equal-momentum quiz on the work–energy page.