Momentum & Impulse

1. Definitions (Must Know)

A. Linear momentum, vector p

Linear momentum, vector p, is a vector defined by:

vector p = m vector v

Unit: kg m s⁻¹ (equivalently N s).

B. Impulse, vector J

Impulse, vector J, is the change in momentum:

vector J = Δ vector p = vector p_f- vector pᵢ

C. Conservation of momentum (principle statement)

If no resultant external force acts on a system (i.e. the system is isolated), total momentum is constant:

∑ vector p_before = ∑ vector p_after

2. Key Ideas (What Earns Marks)

• Momentum is a vector: choose a sign convention (e.g. right is +) and use signs in 1D.
• Impulse equals change in momentum: vector J = Δ vector p.
• On a force–time graph, area = impulse.
• Momentum conservation applies to a system only when external impulse is negligible.
• In a collision, the impulses on the two objects are equal in magnitude and opposite in direction.

3. Detailed Explanations

A. Momentum is directional (signs matter)

Because vector p = m vector v and vector v has direction, momentum has direction.

In 1D collision questions:

1. choose a positive direction,
2. write velocities with signs,
3. conserve momentum.

B. Impulse from a force–time graph (area)

If a force acts for a short time (e.g. a collision), its overall effect is described by impulse:

vector J = Δ vector p

For constant force:

J = FΔ t

For a varying force, use the area under the F–t graph.

The diagram compares equal momentum changes. In a calculation, use signed area: an area below the time axis gives a negative impulse for the chosen positive direction.

C. Why momentum is conserved (system idea)

Consider two objects colliding.

By Newton’s 3rd law, the force on 1 due to 2 equals the opposite of the force on 2 due to 1. Over the same contact time, the impulses are equal and opposite:

Δ vector p₁ = -Δ vector p₂

So the total momentum change is zero:

Δ(vector p₁ + vector p₂) = 0 ⇒ vector pₜₒₜₐₗ = constant

This is why momentum conservation works best when external forces (or external impulse) are negligible during the interaction.

D. Collision workflow (1D, exam-safe)

1. Define a positive direction.
2. Write down p = mv for each object before and after.
3. Apply conservation of momentum: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
4. Add extra information (e.g. “they stick together”, “perfectly elastic”) to close the problem.

See: Elastic & Inelastic Collisions.

4. Common Mistakes

• Conserving momentum for a single object (it’s a system law).
• Dropping direction (signs) in 1D.
• Using F = ma during a collision instead of impulse (F is usually not constant).
• Mixing up “area under F–t” (impulse) with “area under v–t” (displacement).

5. Exam Tips

• Start with: “take rightwards as positive” (or similar).
• Convert units early (g → kg, ms → s).
• If asked for a collision force, use: F_avg = (Δ p)/(Δ t)
• If asked to compare forces during a collision: by Newton’s 3rd law, the forces are equal in magnitude and opposite in direction.

6. Worked Examples

7. Mind Stretchers