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

Newton’s second law can be written in its most general form as

\$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}\$

where \$\mathbf{p}=m\mathbf{v}\$ is the linear momentum of a body. For a constant mass this reduces to the familiar

\$\mathbf{F}_{\text{net}} = m\mathbf{a}\$

Momentum is a vector quantity; its direction is the same as the velocity of the object. The law tells us that any net external force changes the momentum of a system.

Key points

• Momentum is conserved in an isolated system (no external forces).
• Impulse, \$ \mathbf{J} = \int \mathbf{F}\,dt\$, changes the momentum: \$\mathbf{J} = \Delta\mathbf{p}\$.
• Newton’s third law guarantees that internal forces cancel in the total momentum balance.

Frictional Forces – Qualitative Understanding

Friction opposes relative motion (or the tendency for motion) between two surfaces in contact. Two broad categories are relevant at A‑Level:

1. Static friction – prevents the start of motion up to a maximum value.
2. Kinetic (sliding) friction – acts once surfaces are sliding past each other.

Both types arise from microscopic interlocking of surface asperities and from intermolecular forces. The direction of the frictional force is always opposite to the direction of relative motion (or impending motion).

Qualitative features

• Friction does not depend on the area of contact (for typical macroscopic surfaces).
• It is roughly proportional to the normal reaction \$N\$, but the proportionality constant (the coefficient) is not required for this qualitative treatment.
• Friction converts kinetic energy into thermal energy, reducing the speed of moving objects.

Viscous and Drag Forces – Air Resistance

When an object moves through a fluid (air, water, oil) it experiences a resistive force that we call drag. Drag originates from two mechanisms:

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

Both mechanisms act opposite to the direction of motion and increase as the speed of the object increases.

Qualitative behaviour of drag

• At very low speeds (laminar flow) the drag force is approximately proportional to speed: \$F_{\text{drag}} \propto v\$ (Stokes’ regime).
• At higher speeds (turbulent flow) the drag force grows roughly with the square of speed: \$F_{\text{drag}} \propto v^{2}\$.
• The transition between the two regimes occurs around a Reynolds number of order \$10^{3}\$–\$10^{4}\$, but the exact value is not required for this qualitative discussion.

Simple Model of Drag Increasing with Speed

For A‑Level purposes we can adopt a piece‑wise model that captures the essential trend without invoking detailed coefficients:

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

\begin{cases}

k{1}\,v, & v \lesssim v{\text{c}} \\

k{2}\,v^{2}, & v \gtrsim v{\text{c}}

\end{cases}\$\$

where \$k{1}\$ and \$k{2}\$ are positive constants that depend on the shape and size of the object and on the properties of the fluid, and \$v_{\text{c}}\$ is a characteristic speed at which the flow changes from laminar to turbulent.

Key qualitative consequences:

1. When the object is initially at rest, the drag is negligible and the net force is essentially the applied force.
2. As speed builds up, the drag term grows, reducing the net accelerating force.
3. Eventually a terminal speed \$v{\text{t}}\$ is reached when the applied force equals the drag force: \$F{\text{applied}} = F{\text{drag}}(v{\text{t}})\$. At this point the acceleration becomes zero and the object moves at constant speed.

Comparison of Friction and Drag

Summary

• Newton’s second law links net force to the rate of change of momentum.
• Frictional forces oppose relative motion between solid surfaces and are largely speed‑independent.
• Viscous and drag forces oppose motion through fluids; they increase with speed, transitioning from a linear to a quadratic dependence.
• A simple two‑regime model \$F{\text{drag}} \propto v\$ (low \$v\$) and \$F{\text{drag}} \propto v^{2}\$ (high \$v\$) captures the essential physics needed for A‑Level analysis.
• Understanding these resistive forces is essential for correctly applying Newton’s laws to real‑world problems involving motion on surfaces and through air or other fluids.