NDA Mechanics Cheat Sheet - Kinematics, Dynamics & Gravitation Notes

1. Kinematics (Linear & Projectile Motion)

Linear Motion with Constant Acceleration

Kinematics forms the foundational base of general physics in the NDA syllabus. When a body moves under constant acceleration \(a\), its linear motion is described using standard equations of motion:

\[v = v_0 + at\]

\[x = x_0 + v_0t + \frac{1}{2}at^2\]

\[v^2 = v_0^2 + 2a(x - x_0)\]

Here, \(v_0\) represents initial velocity, \(v\) represents final velocity, \(x_0\) is the initial position, \(x\) is the final position, and \(t\) is the time elapsed.

Projectile Motion

A projectile is thrown with an initial velocity \(v_0\) at an angle \(\theta\) with the horizontal. The motion can be decoupled into independent horizontal and vertical components under uniform gravitational acceleration \(g\):

\[x(t) = (v_0 \cos\theta) t\]

\[y(t) = (v_0 \sin\theta) t - \frac{1}{2}gt^2\]

2. Newton's Laws of Motion & Friction

Newton's Laws

• First Law (Law of Inertia): \(\sum \mathbf{F} = 0 \implies \mathbf{a} = 0\). A body remains at rest or moves with uniform velocity unless acted upon by a net external force.
• Second Law: \(\mathbf{F} = \frac{d\mathbf{p}}{dt} = m\mathbf{a}\). Force is the rate of change of linear momentum.
• Third Law: \(\mathbf{A}_{12} = -\mathbf{A}_{21}\). For every action, there is an equal and opposite reaction.

Friction

Frictional force opposes relative motion between contacting surfaces. Static friction \(f_s\) self-adjusts up to its maximum value, whereas kinetic friction \(f_k\) acts during motion:

\[f_s \le \mu_s N, \quad f_k = \mu_k N\]

where \(\mu_s\) is the coefficient of static friction, \(\mu_k\) is the coefficient of kinetic friction, and \(N\) is the normal reaction force.

3. Work, Energy, and Power

Work Done

The work done by a force field along a displacement path \(d\mathbf{r}\) is defined by the line integral:

\[W = \int \mathbf{F} \cdot d\mathbf{r}\]

Kinetic Energy & Potential Energy Relation

Kinetic energy \(K\) of a mass \(m\) moving with velocity \(v\) is given by:

\[K = \frac{1}{2}mv^2\]

For a conservative force field, the force component along a spatial variable \(x\) is related to potential energy \(U\) by:

\[F_x = -\frac{dU}{dx}\]

Work-Energy Theorem & Power

The net work done by all forces equals the change in kinetic energy:

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

Instantaneous power \(P\) is the time rate of doing work or scalar product of force and velocity vectors:

\[P = \mathbf{F} \cdot \mathbf{v} = \frac{dW}{dt}\]

4. Momentum, Impulse & Collisions

Linear Momentum & Impulse

Linear momentum \(\mathbf{p}\) is defined as \(\mathbf{p} = m\mathbf{v}\). Impulse \(\mathbf{J}\) measures the overall effect of a force applied over a time interval:

\[\mathbf{J} = \int \mathbf{F} dt = \Delta \mathbf{p}\]

Law of Conservation of Linear Momentum

If the net external force on a system is zero (\(\sum \mathbf{F}_{ext} = 0\)), total linear momentum is conserved: \(\sum \mathbf{p} = \text{constant}\).

One-Dimensional Elastic Collisions

For two colliding bodies of masses \(m_1\) and \(m_2\) moving with initial velocities \(v_1\) and \(v_2\), the post-collision velocity \(v_1'\) of mass \(m_1\) in a 1D head-on elastic collision is:

\[v_1' = \frac{m_1 - m_2}{m_1 + m_2}v_1 + \frac{2m_2}{m_1 + m_2}v_2\]

5. Rotational Dynamics

Linear vs. Rotational Analogies

Rotational motion around a fixed axis directly parallels linear kinematics and dynamics:

Key Equations of Rotational Motion

• Torque: \(\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} = I\boldsymbol{\alpha}\)
• Rotational Kinetic Energy: \(K_{\text{rot}} = \frac{1}{2}I\omega^2\)
• Angular Momentum: \(\mathbf{L} = I\boldsymbol{\omega} = \mathbf{r} \times \mathbf{p}\)

6. Gravitation & Kepler's Laws

Newton's Law of Universal Gravitation

The attractive force between two point masses \(m_1\) and \(m_2\) separated by distance \(r\) is:

\[F = G \frac{m_1 m_2}{r^2}\]

Gravitational Potential Energy

Taking zero potential at infinity, the gravitational potential energy of two masses is:

\[U = -G \frac{m_1 m_2}{r}\]

Kepler's Third Law of Planetary Motion

The square of the orbital period \(T\) of a planet or satellite is directly proportional to the cube of its orbital radius \(r\):

\[T^2 = \bigl(\frac{4\pi^2}{GM}\bigr) r^3\]

7. Exam Pattern, PYQ Analysis & Chapter Weightage

In the National Defence Academy (NDA) examination conducted by UPSC, General Science Physics accounts for a substantial weightage. Mechanics is consistently the highest-yielding sub-topic within Physics.

Key NDA Mechanics Trends:

• Kinematics & Newton's Laws: Direct numericals and conceptual questions on projectile range, friction limits, and momentum conservation are frequently asked.
• Work-Energy Theorem & Gravitation: Concepts like orbital velocity, escape velocity, time period variations of satellites, and work done by conservative forces appear regularly.
• Difficulty Level: Focuses primarily on Class 11 Physics fundamentals with conceptual applications rather than heavy multi-step calculus.

8. Frequently Asked Questions (FAQ)