Laws of motion

Laws of Motion – IIT‑JEE/Competitive Physics Notes

(Prepared by an expert teacher for AIIMS Admission)

Goal – Master the conceptual and mathematical essence of Newton’s laws, momentum, impulse, and work–energy.
Exam focus – Precise definitions, derivations from first principles, and subtle points that frequently appear in JEE‑Mains, JEE‑Advanced, and AIIMS physics problems.

1. Core Definitions

Notation – Vectors are bold. All derivations assume an inertial frame unless specified.

2. Newton’s Laws – Statement & Consequences

2.1 First Law (Inertia)

In an inertial frame, a particle continues in uniform motion unless acted upon by a net external force.

• Consequence: If (\mathbf{F}_{\text{net}} = \mathbf{0}) → (\mathbf{a} = \mathbf{0}).
• Application: Straight‑line motion, satellite orbits, free‑fall in vacuum.

2.2 Second Law (Fundamental Relation)

The rate of change of linear momentum equals the net external force.

[ \boxed{\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}} ]

• Special case (constant mass): (\mathbf{p} = m\mathbf{v}) → (\mathbf{F}_{\text{net}} = m\mathbf{a}).
• Derivation:
  [ m\mathbf{a} = \frac{d}{dt}(m\mathbf{v}) = \frac{dm}{dt}\mathbf{v} + m\frac{d\mathbf{v}}{dt} ] For constant (m), first term vanishes.

2.3 Third Law (Action–Reaction)

For every action there is an equal and opposite reaction.

[ \mathbf{F}{12} = -\mathbf{F}{21} ]

• Implication: Forces always occur in pairs; no “self‑action”.
• Tricky point: In a single-body analysis, only external forces appear in Newton’s 2nd law.

3. Momentum & Impulse

3.1 Momentum Conservation

• Closed system (no external forces) → (\displaystyle \frac{d\mathbf{p}{\text{tot}}}{dt}=0 \Rightarrow \mathbf{p}{\text{tot}}= \text{constant}).
• Collision problems: Use conservation of momentum + energy (elastic) or just momentum (inelastic).

3.2 Impulse–Momentum Theorem

[ \boxed{\mathbf{I}= \int \mathbf{F},dt = \Delta \mathbf{p}} ]

• Interpretation: The impulse delivered by a force equals the change in momentum.
• Typical exam trick: Compute the impulse from a simple force–time graph (area under curve).

3.3 Work–Energy Theorem (derived from 2nd law)

[ \boxed{W_{\text{net}} = \Delta K} ]

• Derivation:
  [ W_{\text{net}} = \int \mathbf{F}{\text{net}}\cdot d\mathbf{s} = \int m\mathbf{a}\cdot d\mathbf{s} = m\int \frac{d\mathbf{v}}{dt}\cdot d\mathbf{s} = m\int \mathbf{a}\cdot \mathbf{v},dt = \frac{1}{2}m v^2\Big|{i}^{f} ]
• Caveat: Only conservative forces contribute to potential energy considerations; non‑conservative (friction) appear directly in work.

4. Forces in Common Problems