show a qualitative understanding of frictional forces and viscous/drag forces including air resistance (no treatment of the coefficients of friction and viscosity is required, and a simple model of drag force increasing as speed increases is sufficie

Momentum and Newton’s Laws of Motion

Learning Objective

Develop a qualitative understanding of the forces that oppose motion – namely frictional forces and viscous/drag forces (including air resistance). No quantitative treatment of coefficients of friction or viscosity is required; a simple model where drag increases with speed is sufficient.

1. Newton’s Laws – a brief reminder

1. First law (inertia): An object remains at rest or in uniform straight‑line motion unless acted upon by a net external force.
2. Second law: The net force \$\mathbf{F}_{\text{net}}\$ on a body equals the rate of change of its momentum,

   \$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt},\qquad \mathbf{p}=m\mathbf{v}.\$

   For constant mass this reduces to \$\mathbf{F}=m\mathbf{a}\$.

3. Third law: For every action there is an equal and opposite reaction.

2. Momentum – qualitative picture

Momentum \$\mathbf{p}=m\mathbf{v}\$ is a vector quantity that measures the “quantity of motion”. When a force acts, the momentum changes in the direction of the force. The larger the resisting force (e.g., friction or drag), the more rapidly the momentum is reduced.

3. Frictional Forces

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

• Static friction: Prevents motion up to a maximum value; it adjusts itself to match the applied force.
• Kinetic (sliding) friction: Acts once motion has started; its magnitude is roughly constant for a given pair of surfaces.

In the qualitative approach we treat kinetic friction as a constant force \$F_{\text{fr}}\$ acting opposite to the direction of velocity.

4. Viscous and Drag Forces

When an object moves through a fluid (liquid or gas), the fluid exerts a resistive force that depends on the speed of the object.

• Viscous (laminar) drag: At low speeds the fluid layers slide past each other smoothly; the resistive force is approximately proportional to speed,

  \$F{\text{drag}} \approx k1 v,\$

  where \$k_1\$ is a constant that depends on fluid viscosity and object shape.

• Inertial (turbulent) drag: At higher speeds the flow becomes turbulent; the resistive force grows roughly with the square of speed,

  \$F{\text{drag}} \approx k2 v^{2},\$

  where \$k_2\$ reflects fluid density and cross‑sectional area.

5. Air Resistance – a special case of drag

Air resistance is the drag force experienced by objects moving through the atmosphere. For most A‑Level problems a simple model is sufficient:

1. At low speeds (e.g., a falling leaf) use \$F_{\text{air}} \propto v\$.
2. At higher speeds (e.g., a cyclist or a sky‑diver) use \$F_{\text{air}} \propto v^{2}\$.

The direction of the air‑resistance force is always opposite to the instantaneous velocity vector.

6. Simple Drag Model for Qualitative Analysis

When a problem requires only a qualitative description, adopt the following piece‑wise model:

\$\$F_{\text{drag}}(v)=

\begin{cases}

k1 v, & v < v{\text{c}}\\[4pt]

k2 v^{2}, & v \ge v{\text{c}}

\end{cases}\$\$

Here \$v{\text{c}}\$ is a characteristic speed at which the flow changes from laminar to turbulent. The exact values of \$k1\$, \$k2\$, and \$v{\text{c}}\$ are not required for a qualitative discussion; what matters is the trend that the resisting force grows as speed increases.

7. Comparison of Forces

8. How the Forces Affect Momentum

From Newton’s second law, a resistive force \$\mathbf{F}_{\text{res}}\$ changes the momentum according to

\$\frac{d\mathbf{p}}{dt} = -\mathbf{F}_{\text{res}}.\$

Because \$\mathbf{F}_{\text{res}}\$ always points opposite to \$\mathbf{v}\$, the magnitude of momentum decreases. The rate of decrease is faster when the resistive force grows more steeply with speed (e.g., \$v^{2}\$ drag).

9. Example Scenarios (Qualitative)

• Block sliding on a rough floor: Kinetic friction provides a constant deceleration; the block’s speed reduces linearly with time.
• Sphere falling through a viscous liquid: The drag force \$F\propto v\$ leads to an exponential approach to a terminal speed where \$mg = k1 v{\text{t}}\$.
• Sky‑diver after the jump: At early times \$v\$ is small, so \$F{\text{air}}\approx k1 v\$; later, \$v\$ becomes large and \$F{\text{air}}\approx k2 v^{2}\$, giving a lower terminal speed than in the linear regime.

10. Summary

• Newton’s laws link forces to changes in momentum.

• Frictional forces act between solid surfaces; kinetic friction is roughly constant.

• Viscous/drag forces arise from motion through fluids; they increase with speed – linearly at low speeds, quadratically at high speeds.

• Air resistance is a specific drag force; a simple piece‑wise model captures its qualitative behaviour.

• Understanding how these forces scale with speed helps predict how quickly an object’s momentum will be reduced.