Mechanics - Motion, Forces and Energy

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s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time. The five equations: v = u + at. s = ut + ½at². s = vt − ½at². v² = u² + 2as. s = ½(u+v)t. Used when acceleration is constant.

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An object remains at rest or in uniform motion in a straight line unless acted upon by an unbalanced (net) external force. This is the law of inertia. Example: a spacecraft in deep space continues moving at constant velocity with no engines.

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F = ma — the net force on an object equals its mass times its acceleration. Alternatively: Force = rate of change of momentum (F = Δp/Δt). If mass is constant: F = ma. Units: Newtons (N = kg·m/s²).

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For every action, there is an equal and opposite reaction. When object A exerts a force on object B, object B simultaneously exerts an equal and opposite force on object A. The forces act on different objects — they never cancel each other out.

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p = mv (momentum = mass × velocity). A vector quantity — has direction. Units: kg·m/s or N·s. Conservation of momentum: In a closed system, total momentum before = total momentum after a collision.

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Impulse = F × Δt = Δp (change in momentum). A large force applied for a short time produces the same impulse as a small force applied for a longer time. Explains why airbags and crumple zones save lives — they increase collision time → reduce peak force.

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Elastic: Both momentum AND kinetic energy are conserved. Objects bounce off perfectly — no deformation. Ideal (rare). Inelastic: Momentum is conserved but kinetic energy is NOT (converted to heat, sound, deformation). Real collisions are inelastic. Perfectly inelastic: Objects stick together — maximum KE loss.

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GPE = mgh where m = mass (kg), g = gravitational field strength (9.81 m/s² on Earth), h = height above reference point (m). Units: Joules (J). GPE increases as height increases.

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KE = ½mv² where m = mass (kg), v = velocity (m/s). Units: Joules (J). KE is proportional to the square of velocity — doubling speed quadruples KE.

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Energy cannot be created or destroyed — only converted from one form to another. The total energy of an isolated system remains constant. Example: GPE converts to KE as an object falls: mgh = ½mv² (ignoring air resistance).

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W = Fd cos θ where F = force (N), d = displacement (m), θ = angle between force and displacement. Work is done only when the force has a component in the direction of motion. Units: Joules (J). No work is done if displacement is zero or force is perpendicular to motion.

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P = W/t = Fv where P = power (W), W = work done (J), t = time (s), F = force, v = velocity. Power is the rate of doing work or transferring energy. Units: Watts (W = J/s).

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Efficiency = (useful energy output / total energy input) × 100%. No real machine is 100% efficient — energy is always lost (usually as heat). Efficiency < 1 (or < 100%). Improving efficiency reduces energy losses.

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The curved path followed by an object launched into the air under gravity alone (no air resistance). Horizontal: constant velocity (no horizontal force). Vertical: constant downward acceleration (g = 9.81 m/s²). The two components are independent — treat separately.

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An object moving in a circle at constant speed is constantly changing direction → it is accelerating (centripetal acceleration toward centre). Centripetal force F = mv²/r = mω²r — always directed toward the centre. Provided by: gravity (planets), tension (string), normal force (car rounding a bend), friction (tyres).

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a = v²/r = ω²r — directed toward the centre of the circular path. Even at constant speed, an object in circular motion is accelerating because its direction is changing. The centripetal force is what produces this centripetal acceleration.

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Oscillatory motion where the restoring force is proportional to and directed opposite to the displacement from equilibrium: F = −kx. Examples: mass-spring system, simple pendulum (small angles). Characteristics: sinusoidal displacement, constant amplitude (no damping), fixed period.

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T = 2π√(L/g) where L = length (m), g = gravitational field strength (m/s²). Period is independent of mass and amplitude (for small angles). Increasing length increases period. Used in clocks.

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A force opposing relative motion between surfaces. Static friction: Prevents motion from starting — can vary up to a maximum value. Kinetic (sliding) friction: Acts when surfaces are sliding — usually constant and less than maximum static friction. f = μN where μ = coefficient of friction, N = normal force.

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Mass: The amount of matter in an object — scalar, measured in kg, constant everywhere. Weight: The gravitational force on an object — vector, measured in Newtons. W = mg where g varies by location (9.81 m/s² on Earth, 1.62 m/s² on the Moon).

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For a body in rotational equilibrium: Sum of clockwise moments = Sum of anticlockwise moments. Moment = Force × perpendicular distance from pivot. Units: N·m. Used to solve problems with levers, beams, and seesaws.

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The force needed to extend or compress a spring by some distance is proportional to that distance (within the elastic limit): F = kx where k = spring constant (N/m) and x = extension or compression (m). Beyond the elastic limit, the spring is permanently deformed.

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The constant maximum velocity reached when the drag force equals the weight of the object — net force = 0, so acceleration = 0. Factors: heavier objects have higher terminal velocity; streamlined shapes reduce drag → higher terminal velocity. Parachute greatly increases drag → very low terminal velocity.

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The force per unit mass at a point in a gravitational field: g = F/m (N/kg or m/s²). On Earth's surface: g ≈ 9.81 m/s². Decreases with distance from Earth (g ∝ 1/r²). At the Moon's surface: ~1.62 m/s². In space far from any body: ≈ 0.

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Every mass attracts every other mass: F = Gm₁m₂/r² where G = 6.67 × 10⁻¹¹ N·m²/kg² (universal gravitational constant), m₁ and m₂ = masses, r = distance between centres. Force decreases as inverse square of distance. Acts along the line joining the centres.