Momentum: conserved motion with direction

AQA A Level Physics (3.4.1.6)
Lower Sixth • 35 minutes

Learning objectives

1. Define momentum: $\mathbf{p}=m\mathbf{v}$ (vector; units).
2. Apply conservation of momentum in 1D for an isolated/approximately isolated system.
3. Use $F=\Delta p/\Delta t$ and interpret impulse as area under an $F$-$t$ graph.
4. Distinguish momentum conservation from kinetic-energy conservation.

Key vocabulary

• vector (magnitude + direction), scalar (magnitude only)
• system, external force, internal force
• impulse, elastic, inelastic

What you already know vs what is new today

Already covered (3.4.1.1—3.4.1.5 and 3.4.1.7—3.4.1.8)

• Core mechanics language: vector/scalar, components, sign conventions, units.
• Kinematics basics: $v$, $a$, and interpreting velocity–time graphs (including “area under graph” reasoning).
• Rate-chain recap: $\Delta x$ over time gives $v$, and $\Delta v$ over time gives $a$.
  For interested students only: $\Delta a$ over time is jerk (not required for this lesson/spec point).
• Newton’s laws, especially $F=ma$ for constant mass and free-body diagrams. ([aqa.org.uk][1])
• Work/energy: $E_k=\tfrac12 mv^2$, $W=Fs\cos\theta$, power ideas. ([aqa.org.uk][1])

New in 3.4.1.6 (today)

• Momentum defined and used as a conserved quantity.
• Impulse and force as rate of change of momentum $\left(F=\Delta(mv)/\Delta t\right)$, and for constant $F$, $F\Delta t=\Delta p$.
• Force–time graphs: area under the curve as impulse.
• Applying conservation of momentum quantitatively in 1D; elastic vs inelastic; explosions; “impact time reduces force” contexts. ([aqa.org.uk][1])

Introduction: symmetry suggests which quantities matter

Reversing direction changes $v \to -v$.
But squaring removes direction: $(-v)^2=v^2$.

So it is natural that:

• momentum is linear in $v$ because it tracks direction;
• kinetic energy is proportional to $v^2$ because it ignores direction.

Symmetry helps justify why energy uses $v^2$.
The factor $\tfrac12$ is not explained by symmetry; it comes from geometry of areas under graphs, later.

Textbook core 1: momentum and mass

Momentum

[ \mathbf{p}=m\mathbf{v} ]

• Units: $\mathrm{kg\,m\,s^{-1}}$
• Momentum is a vector (direction matters).

1D sign convention: choose a positive direction and use signs for velocities.

Mass as proportionality (and as “quantity of matter”)

In this topic we can treat mass as the constant of proportionality between $p$ and $v$:

[ p \propto v \quad\Rightarrow\quad p = mv. ] Experiments with ordinary matter show something stronger: mass is also (very nearly) a conserved property of matter in everyday mechanics, used as a measure of “quantity of matter”.

Aside: directional motion is vector motion, whereas “amount” without direction is scalar motion.

Textbook core 2: conservation of momentum

Conservation law (1D)

If the resultant external force on a system is zero (or external impulse is negligible during the interaction),

[ \sum p_{\text{before}} = \sum p_{\text{after}}. ]

What “system” means

A system is the collection of objects you decide to include in the momentum balance (e.g., two trolleys together).

• Internal forces act between objects inside the system (they come in equal and opposite pairs).
• External forces come from outside the system (track friction, a hand pushing, gravity components along the track).

Momentum conservation is strongest when external forces are negligible during the short interaction.

Textbook core 3: force, impulse, and graphs

Newton’s second law in momentum form

[ F=\frac{dp}{dt}. ] Definition: Impulse is the change in momentum,

[ \text{Impulse} \equiv \Delta p. ] For (approximately) constant force over $\Delta t$,

[ \Delta p = F\Delta t. ] More generally,

[ \Delta p=\int F,dt. ]

Graph meaning (exam gold)

• On an $F$-$t$ graph, area $=\Delta p$ (impulse).
• On a $p$-$t$ graph, gradient $=\dfrac{dp}{dt}=F$.

What we will do in 35 minutes (lesson spine)

1. Warm-up diagnostic (3–4 min): quick questions to activate prior knowledge.
2. Define momentum (5 min): $p=mv$, vector and units; why it’s different from energy.
3. Conservation in 1D (12–15 min): one model example + pupil practice (stick then rebound).
4. Impulse (8–10 min): $F=\Delta p/\Delta t$; force–time area; “increase impact time, reduce force”.
5. Exit check (2 min): one conservation question + one impulse-from-graph question.

Warm-up diagnostic questions (3—4 minutes)

1. A $2\,\text{kg}$ object at $+3\,\text{m s}^{-1}$: what is $E_k$?
2. If the resultant force is zero, what can you say about velocity?
3. Sketch what a large force over a short time might look like on an $F$-$t$ graph.

Board plan (teacher script, left → right boards)

Figures (caption immediately after each image)

Experimental results (demo table)

Interpretation: agreement won’t be perfect (friction, alignment, sensor noise). “Close” is evidence for the conservation model during the short interaction.

Textbook fill: standard collision outcomes

Inelastic vs elastic

• Inelastic collision: momentum conserved (if isolated), kinetic energy decreases (converted to heat/sound/deformation).
• Perfectly inelastic: objects stick together (maximum kinetic energy loss consistent with momentum conservation).
• Elastic collision: momentum conserved and kinetic energy conserved (idealised; approximately true in some interactions).

Important: momentum conservation depends on external forces, not on whether the collision is elastic. ([aqa.org.uk][1])

Explosions (useful extension)

If a system starts at rest and explodes into two parts, total momentum is still zero:

[ p_1 + p_2 = 0 \quad\Rightarrow\quad p_1 = -p_2. ] This is a clean “direction matters” example.

Textbook fill: common mistakes (and how to avoid them)

1. Forgetting direction: always set a positive direction and keep signs.
2. Conserving momentum for the wrong system: include all interacting objects; exclude the hand pushing.
3. Mixing up graphs:
   • $F$-$t$: area is $\Delta p$.
   • $p$-$t$: gradient is $F$.
4. Assuming kinetic energy is conserved: only in elastic collisions.

Questions to test understanding

Quick checks

1. Why is momentum a vector? Give a 1D example where the sign matters.
2. Show dimensionally: $\mathrm{N}=\mathrm{kg\,m\,s^{-2}}$ and $dp/dt$ has the same units.
3. What assumption are you making when you apply momentum conservation to two trolleys on a track?
4. On an $F$-$t$ graph, what does the area represent?

Short problems

5. Stick collision: $m_1=0.50$ kg at $+1.2$ m/s hits $m_2=0.50$ kg at rest and they stick. Find $v$.
6. Rebound: $m_1=0.40$ kg at $+3.0$ m/s hits $m_2=0.60$ kg at rest. After, $v_1=-1.0$ m/s. Find $v_2$.
7. Impulse: approximate a triangular force pulse with $F_{\max}=120\ \mathrm{N}$ lasting $0.040\ \mathrm{s}$. Estimate $\Delta p$.

Conceptual

8. Two pulses have the same area but different peaks. What is the same physically? What differs?
9. Why does increasing collision time reduce injury risk if $\Delta p$ is fixed?

Practical/apparatus notes (low risk, high payoff)

If you want apparatus, the most interview-efficient is something that makes the collision story concrete without setup risk:

• two low-friction trolleys with Velcro/magnets for “stick together” collisions; and/or
• a motion sensor/light gates to measure before/after velocities.

If the lesson is fully runnable without apparatus, the demo becomes an enrichment/evidence moment rather than a dependency.

Interactive trolley simulation

Further thoughts (beyond syllabus): curved $p$-$v$ and “effective mass”

What could it represent? (suggestions)

• A system gaining/losing mass as it moves (e.g. collecting material).
• A situation where the simple model “one constant $m$” isn’t adequate across the whole range (you would test what extra physics is missing).

Thinking exercises

1. If the curve bends upward (slope increasing with $v$), what does that suggest about $m(v)$?
2. How would you estimate the local slope near $v=v_0$ using a small $\Delta p$–$\Delta v$ triangle?
3. Why is it risky to label the shaded area “kinetic energy” in the variable-mass case?
4. What extra measurements would you want to decide whether curvature is truly “mass changing” or a different effect?

Plenary (syllabus)

• $\mathbf{p}=m\mathbf{v}$ (vector; sign convention in 1D).
• If external impulse is negligible: $\sum p_{\text{before}}=\sum p_{\text{after}}$.
• $\Delta p=\int F\,dt$ (area under $F$-$t$).
• $E_k=\tfrac12 mv^2$ is not always conserved even when momentum is.