1.6: MOMENTUM

CORE VS EXTENDED GUIDE

• 📌 Core students study only the unlabelled sections.

• 📌 Extended students must study everything, including 🚀 Extended points.

• 📌 Extended = Core + Supplement.

🚀 Momentum

Key Equations

Momentum:

$$ p = mv $$

Variables:

• $p$ = momentum, in kg m s⁻¹ (or N s)

• $m$ = mass, in kg

• $v$ = velocity, in m s⁻¹

SI unit of momentum: kg m s⁻¹

Rearrangements:

$$ m = \frac{p}{v} \qquad v = \frac{p}{m} $$

Proportionality: Momentum is directly proportional to mass when velocity is constant. Momentum is directly proportional to velocity when mass is constant. Therefore, doubling either mass or velocity doubles the momentum.

📌 Momentum is the product of mass and velocity.

Momentum describes how much "motion content" an object carries — a heavy lorry at low speed can have the same momentum as a light car at high speed. Because velocity is a vector quantity, momentum is also a vector: its direction is the same as the direction of the object's velocity. This means two objects moving in opposite directions have momenta [plural of momentum] with opposite signs.

Sign convention for one-dimensional problems: choose one direction as positive (e.g. rightward = positive). Any object moving in the opposite direction has a negative velocity, and therefore a negative momentum.

MisconceptionStudents sometimes treat momentum as a scalar and ignore direction. Momentum is a vector. In every calculation, assign a positive direction first and use negative signs for the opposite direction. Exam cue: always state "Taking [direction] as positive" before substituting values.

🚀 Impulse

Key Equations

Impulse:

$$ \text{impulse} = F \Delta t = \Delta(mv) $$

Variables:

• $F$ = force, in N

• $\Delta t$ = time for which the force acts, in s

• $\Delta(mv)$ = change in momentum, in kg m s⁻¹

SI unit of impulse: N s (equivalent to kg m s⁻¹)

Rearrangements:

$$ F = \frac{\Delta(mv)}{\Delta t} \qquad \Delta t = \frac{\Delta(mv)}{F} $$

Proportionality: Impulse is directly proportional to the force when time is constant. Impulse is directly proportional to the time when force is constant. Doubling the force doubles the impulse (for the same time interval).

📌 Impulse is the product of force and the time for which the force acts.

Impulse equals the change in momentum of the object. A large force acting for a short time can produce the same change in momentum as a small force acting for a longer time — this is why car crumple zones work. The crumple zone increases the collision time $\Delta t$, so for the same change in momentum, the force $F$ on the passengers decreases.

Examiner InsightCIE commonly asks "explain how a safety feature reduces injury." The required chain is: the feature increases the time of impact → for the same change in momentum, impulse = FΔt → the force is reduced → less injury. Exam cue: always link the longer time to a smaller force via the impulse equation.

🚀 Conservation of Momentum

📌 The principle of the conservation of momentum states that the total momentum of a system of objects remains constant, provided no resultant external force acts on the system.

This means that in any collision or explosion, the total momentum before the event equals the total momentum after the event:

$$ \text{total momentum before} = \text{total momentum after} $$

$$ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 $$

where $u$ represents velocity before and $v$ represents velocity after.

This principle applies to all types of interaction in one dimension:

For an explosion from rest, the total initial momentum is zero. Therefore the total final momentum must also be zero, meaning the momenta of the fragments are equal in magnitude but opposite in direction.

MisconceptionStudents often think momentum is not conserved when objects stop after a collision. If both objects stop, the total momentum after is zero — which means the total momentum before was also zero (they were moving in opposite directions with equal-magnitude momenta). Exam cue: momentum is always conserved in the absence of external forces, regardless of what happens to kinetic energy.

Worked Example — Explosion

A 3.0 kg cannon fires a 0.020 kg ball at 150 m s⁻¹. Calculate the recoil velocity of the cannon.

Taking the direction of the ball as positive. Total momentum before firing = 0 (both at rest).

Finding the recoil velocity

$$ \text{total momentum before} = \text{total momentum after} $$

$$ 0 = m_{\text{ball}} v_{\text{ball}} + m_{\text{cannon}} v_{\text{cannon}} $$

Substitution:

$$ 0 = (0.020 \times 150) + (3.0 \times v_{\text{cannon}}) $$

$$ 0 = 3.0 + 3.0 \times v_{\text{cannon}} $$

Rearranging for $v_{\text{cannon}}$:

$$ v_{\text{cannon}} = \frac{-3.0}{3.0} $$

$$ v_{\text{cannon}} = -1.0 \text{ m s}^{-1} $$

The negative sign indicates the cannon recoils in the opposite direction to the ball, as expected.

Exam Tipensure arrow directions match the sign convention chosen, and label all masses and velocities with values.

🚀 Resultant Force and Rate of Change of Momentum

Key Equations

Resultant force (Newton's second law in momentum form):

$$ F = \frac{\Delta p}{\Delta t} $$

Variables:

• $F$ = resultant force, in N

• $\Delta p$ = change in momentum, in kg m s⁻¹

• $\Delta t$ = time interval, in s

SI unit: N (where 1 N = 1 kg m s⁻² = 1 kg m s⁻¹ per s)

Rearrangements:

$$ \Delta p = F \Delta t \qquad \Delta t = \frac{\Delta p}{F} $$

Proportionality: Resultant force is directly proportional to the change in momentum when time is constant. Resultant force is inversely proportional to the time interval when the change in momentum is constant. Doubling the time over which the same momentum change occurs halves the force.

📌 Resultant force is the change in momentum per unit time.

This equation is the most general form of Newton's second law. When mass is constant, $F = \frac{\Delta(mv)}{\Delta t}$ simplifies to $F = m \times \frac{\Delta v}{\Delta t} = ma$. However, the momentum form is more powerful because it also applies to situations where mass changes (such as a rocket burning fuel), and it directly links force to impulse.

The direction of the resultant force is the same as the direction of the change in momentum. If no resultant force acts, $\Delta p = 0$, so momentum is conserved — this connects directly back to the conservation of momentum principle.

Examiner InsightCIE may give a momentum--time graph and ask for the resultant force. The gradient of a momentum--time graph equals the resultant force, because gradient = $\frac{\Delta p}{\Delta t} = F$. A steeper gradient means a larger force. Exam cue: if given a p--t graph, calculate the gradient to find the force.

Exam Tipread the gradient carefully — use Δp ÷ Δt from two widely separated points on the line for accuracy.