PA02 — Mechanics: Motion, Force, Work & Power

Uniform motion: constant speed in a straight line (constant velocity). Uniform circular motion has constant speed but changing direction — it is NOT uniform motion. Uniformly accelerated motion has changing speed. Non-uniform motion = variable speed.

Newton's First Law: a body remains at rest or in uniform motion unless acted upon by an external force. This is the law of inertia. Inertia is the tendency of a body to resist changes in its state of motion. Mass is the measure of inertia.

Momentum (p) = mass × velocity = kg × m/s = kg·m/s. 1 N·s = 1 kg·m/s (since 1 N = 1 kg·m/s²). Both C and D are correct — but Newton-second is the standard derived unit name. Options C and D are equivalent; the question tests recognising both forms.

At the highest point of vertical projection: velocity = 0 (momentarily at rest before falling back). Acceleration = g = 9.8 m/s² downward (gravity always acts downward — it never becomes zero). This is a very commonly tested concept in AFCAT.

Work = F × d × cos θ. When θ = 90° (force perpendicular to displacement), cos 90° = 0 → W = 0. Example: a centripetal force in circular motion does zero work because it is always perpendicular to velocity. Normal reaction from a horizontal surface does zero work on a horizontally moving object.

Speed is a scalar (magnitude only, no direction). Velocity, force, and displacement are vectors (they have both magnitude and direction). Scalar examples: speed, distance, mass, time, energy, temperature, density.

Conservation of momentum: total momentum remains constant when no net external force acts on the system. This holds regardless of internal forces (they cancel in pairs by Newton's 3rd law). Applications: rocket propulsion, recoil of guns, collisions. Momentum is always conserved in collisions.

Uniform circular motion: speed is constant but velocity direction changes continuously. This change in velocity direction constitutes centripetal acceleration (a = v²/r) directed toward the centre. There is no tangential acceleration (speed doesn't change). Centripetal force = mv²/r provides this acceleration.

Power = Work / Time = Energy / Time. SI unit = Watt (W) = J/s. Equivalently, Power = Force × velocity (when F and v are parallel). 1 HP (horsepower) = 746 W. Power is a scalar quantity with dimensions [ML²T⁻³].

Range R = u²sin(2θ)/g. For maximum range, sin(2θ) = 1 → 2θ = 90° → θ = 45°. At 45°, both horizontal and vertical components of initial velocity are equal, giving maximum range. At 90° (vertical throw), range = 0. At 30° and 60°, the range is the same (sin 60° = sin 120°) but less than at 45°.

Newton's 2nd Law: F = ma → a = F/m = 10/5 = 2 m/s². This is the most direct application of Newton's Second Law. Force in Newtons, mass in kg → acceleration in m/s².

Centripetal force is the net inward force directed toward the centre of the circular path — it is required to maintain circular motion. It is not a separate new force but is provided by existing forces (tension, friction, gravity, normal force, etc.). It causes centripetal acceleration v²/r.

Elastic potential energy = ½kx², where k = spring constant and x = compression/extension. This follows from Hooke's Law (F = kx) and integration of work done: PE = ∫kx dx from 0 to x = ½kx². Doubling x quadruples the stored energy.

Banking (tilting) of curved roads provides the required centripetal force through the horizontal component of the normal reaction, reducing reliance on friction. Ideal banking angle: tan θ = v²/rg. This allows vehicles to negotiate curves safely even on wet/icy roads.

Escape velocity from Earth = √(2gR) = √(2 × 9.8 × 6.4×10⁶) ≈ 11.2 km/s. This is the minimum speed needed to escape Earth's gravitational field. Orbital velocity (7.9 km/s) is needed for circular orbit. Note: escape velocity = √2 × orbital velocity.

The gun recoils (kicks back) due to conservation of momentum. Before firing: total momentum = 0. After firing: bullet moves forward (positive momentum) → gun moves backward (equal and opposite momentum) so total remains 0. This is Newton's 3rd law in action.

At the midpoint of its trajectory, a vertically thrown ball has fallen (or risen) half the maximum height. Using energy conservation: total E = mgh (at max height, all PE). At midpoint (h/2): PE = mg(h/2), KE = mgh - mg(h/2) = mg(h/2). So KE:PE = 1:1 at the midpoint.

When the string breaks, the centripetal force vanishes. The stone continues in the direction of its instantaneous velocity (tangent to the circle) due to Newton's First Law. This tangential direction is perpendicular to the radius at that point.

Work-Energy Theorem: Net work done on a body = change in kinetic energy (W_net = ΔKE = ½mv² - ½mu²). This applies to any type of force — conservative or non-conservative. It is derived directly from Newton's Second Law by integration.

Work against gravity = mgh = 10 × 10 × 5 = 500 J. This work is stored as gravitational potential energy. W = mgh applies when height h is small compared to Earth's radius (uniform gravity assumed). Work done by gravity = -500 J (negative as gravity opposes upward displacement).