Cambridge Notes, Past Papers, Revision Questions

Dynamics – Momentum, Newton’s Laws, Friction & Drag (Cambridge 9702)

1. Newton’s Three Laws (statement only)

Newton I – Law of Inertia: A body remains at rest or moves with constant velocity unless acted on by a net external force.

Newton II – Law of Motion: The net external force on a body is equal to the mass times its acceleration
\$\mathbf{F}_{\text{net}} = m\mathbf{a}.\$
(The more general form $\mathbf{F}_{\text{net}} = d\mathbf{p}/dt$ is useful for deeper study but is not required by the syllabus.)

Newton III – Action – Reaction: For every action force there is an equal and opposite reaction force acting on a different body. Internal forces cancel when analysing the total momentum of an isolated system.

2. Linear Momentum

Momentum is a vector defined by

\$\mathbf{p}=m\mathbf{v}\$

Its direction is the same as the velocity of the object.

2.1 Momentum in two dimensions

Write the momentum vector in components:
\$\mathbf{p}=m\bigl(vx\hat{\mathbf{i}}+vy\hat{\mathbf{j}}\bigr).\$

Conservation of momentum applies to each component separately:
\$\$\sum p{x,\text{initial}}=\sum p{x,\text{final}},\qquad
\sum p{y,\text{initial}}=\sum p{y,\text{final}}.\$\$

Useful for collisions on smooth tables, projectile impacts, etc.

2.2 Impulse

The impulse delivered by a force $\mathbf{F}(t)$ over a time interval $\Delta t$ is

\$\mathbf{J}= \int{t1}^{t_2}\mathbf{F}\,dt = \Delta\mathbf{p}.\$

Impulse changes the momentum of a body without needing the detailed force history.

2.3 Collisions – key ideas

Momentum is always conserved (elastic or inelastic).
\$\mathbf{p}{\text{initial}}=\mathbf{p}{\text{final}}.\$

Elastic collisions (perfectly elastic) also conserve kinetic energy.
\$\$\tfrac12 m1v1^{2}+\tfrac12 m2v2^{2}=
\tfrac12 m1v1'^{2}+\tfrac12 m2v2'^{2}.\$\$
Note: the syllabus does not require the coefficient of restitution; the kinetic‑energy equation is needed only for perfectly elastic collisions.

Inelastic (sticky) collisions – the bodies stick together after impact:
\$v'=\frac{m1v1+m2v2}{m1+m2}.\$

2.4 Example – head‑on collision on a friction‑less track

Cart A: $mA=2.0\;\text{kg},\;vA=+3.0\;\text{m s}^{-1}$ (right).

Cart B: $mB=3.0\;\text{kg},\;vB=-2.0\;\text{m s}^{-1}$ (left).

Find the velocities after a perfectly elastic collision.

Conserve momentum (right positive):
\$mAvA+mBvB = mAvA'+mBvB'.\$

Conserve kinetic energy (elastic case). Solve the two equations simultaneously to obtain
\$\$v_A' = -0.2\;\text{m s}^{-1},\qquad
v_B' = 2.8\;\text{m s}^{-1}.\$\$

The lighter cart rebounds, the heavier cart continues forward faster – a classic elastic‑collision outcome.

3. Frictional Forces – Qualitative Understanding

3.1 What is friction?

Friction opposes relative motion (or the tendency for motion) between two solid surfaces in contact.

Originates from microscopic interlocking of surface asperities and intermolecular attractions.

Direction: always opposite to the direction of relative motion (or impending motion).

3.2 Types of friction

Static friction – prevents the start of motion up to a maximum value.

Kinetic (sliding) friction – acts once the surfaces are sliding past each other.

3.3 Qualitative features (syllabus focus)

Magnitude is roughly proportional to the normal reaction $N$ (the “weight” of the object on the surface).

Independent of the apparent area of contact.

For most macroscopic surfaces the kinetic friction force is essentially independent of speed.

Friction converts mechanical energy into thermal energy, slowing the object.

3.4 Optional quantitative reminder (not required for 9702)

For completeness, the usual expressions are

\$\$F{\text{static}}^{\max}= \mus N,\qquad

F{\text{kinetic}} = \muk N,\$\$

where $\mus$ and $\muk$ are the static and kinetic coefficients of friction. These are useful for extension work but are not examined.

3.5 Illustrative diagram

Block on an inclined plane showing weight components, normal reaction and friction — Block on an inclined plane: weight $mg$ resolved into components, normal reaction $N$, and friction $F_f$ acting up the plane.

4. Viscous/Drag Forces – Qualitative Understanding of Air Resistance

4.1 What is drag?

When an object moves through a fluid (air, water, oil) it experiences a resistive force called drag.

Drag always acts opposite to the direction of motion.

Two main mechanisms:
- Viscous (frictional) drag – due to the fluid’s internal viscosity; dominant at low speeds and for smooth, small objects.
- Pressure (form) drag – caused by pressure differences between the front and rear of the object; becomes important at higher speeds and for bluff bodies.

4.2 Qualitative behaviour of drag

Low‑speed (laminar) regime: drag increases roughly in proportion to speed.
Conceptual form: $F_{\text{drag}}\propto v$.

High‑speed (turbulent) regime: drag increases roughly with the square of speed.
Conceptual form: $F_{\text{drag}}\propto v^{2}$.

The transition occurs when the flow changes from laminar to turbulent (characterised by a Reynolds number of order $10^{3}\!-\!10^{4}$; the exact value is not required for the exam).

Because drag grows with speed, an object subject to a constant applied force eventually reaches a terminal speed $v_t$ where the applied force balances the drag, giving zero net acceleration.

4.3 Example – terminal speed of a falling sky‑diver

Assume a sky‑diver of mass $m=80\;\text{kg}$ reaches a terminal speed when the weight $mg$ is balanced by a quadratic drag force $F{\text{drag}}=k v^{2}$. If the measured terminal speed is $vt=55\;\text{m s}^{-1}$, the drag constant is

\$k = \frac{mg}{v_t^{2}} = \frac{80\times9.8}{55^{2}}\approx 0.026\;\text{kg m}^{-1}.\$

This illustrates how the qualitative idea of “drag ∝ $v^{2}$ at high speed” leads directly to a terminal‑velocity calculation.

4.5 Illustrative diagram – drag on a sphere

Sphere moving through air with drag arrows opposite to velocity — Sphere moving through air: velocity $\mathbf{v}$ to the right, drag force $\mathbf{F}_{\text{drag}}$ to the left.

5. Comparison of Friction and Drag

Aspect	Friction (solid–solid)	Viscous/Drag (solid–fluid)
Typical speed dependence	Nearly independent of speed (static/kinetic friction)	Increases with speed – roughly linear at low $v$, quadratic at high $v$
Direction of force	Opposite to relative motion (or impending motion)	Opposite to the object's velocity
Energy dissipation	Converted to heat at the contact surface	Converted to heat in the fluid and, at high speed, to turbulent kinetic energy
Dependence on normal reaction	Approximately proportional to the normal reaction $N$	Independent of $N$; depends on fluid density, object shape and speed
Typical syllabus‑level formula	$F_f \approx \mu N$ (qualitative)	$F_{\text{drag}}$ ∝ $v$ (low speed) or ∝ $v^{2}$ (high speed)

6. Summary – What You Must Remember for Cambridge 9702

Newton’s three laws are the foundation of dynamics; for the exam you need the simple form $\mathbf{F}=m\mathbf{a}$.

Linear momentum $\mathbf{p}=m\mathbf{v}$ is conserved in isolated systems; treat each component separately in two‑dimensional problems.

Impulse $\mathbf{J}=\Delta\mathbf{p}$ is a convenient way to handle forces acting over short time intervals.

Collisions:
- Momentum is always conserved.
- For perfectly elastic collisions kinetic energy is also conserved (optional depth).
- For perfectly inelastic (sticky) collisions use the combined‑mass formula.

Friction (solid–solid):
- Static and kinetic types; magnitude roughly $\propto N$.
- Direction opposite to relative motion; essentially speed‑independent.
- Energy is dissipated as heat.

Viscous/drag forces (solid–fluid):
- Oppose motion through a fluid.
- Linear with speed at low $v$ (laminar regime), quadratic at high $v$ (turbulent regime).
- Lead to a terminal speed when drag balances the applied force.

Understanding these resistive forces lets you apply Newton’s laws to real‑world situations such as:
- Blocks sliding on surfaces,
- Cars braking on wet roads,
- Sky‑divers reaching terminal velocity,
- Objects falling through air or water.