PC02 — Kinematics & Laws of Motion

Uniform velocity means constant speed AND constant direction. If velocity is constant, its rate of change (acceleration) is zero. Uniform circular motion has constant speed but changing direction — it has centripetal acceleration, so it is not uniform velocity.

Slope of v-t graph = Δv/Δt = acceleration. Area under v-t graph = displacement. Slope of s-t graph = velocity. Area under a-t graph = change in velocity. These graphical interpretations are frequently tested in CDS Physics.

Using v² = u² + 2as: u = 0 (dropped), a = g, s = h. So v² = 2gh → v = √(2gh). This is derived from energy conservation: mgh = ½mv² → v = √(2gh). The mass cancels — all objects fall at the same rate regardless of mass.

Newton's 3rd Law: every action has an equal and opposite reaction — but these forces act on different bodies (not on the same body, so they cannot cancel each other). Examples: walking (foot pushes ground back, ground pushes foot forward), rocket propulsion, swimming.

R = u²sin(2θ)/g. Maximum when sin(2θ) = 1, i.e., 2θ = 90°, θ = 45°. The range at 30° equals that at 60° (complementary angles give same range). At 90°, range = 0 (vertical throw). At 45°, horizontal and vertical components of velocity are equal.

The three standard equations of uniform motion are: v = u + at; s = ut + ½at²; v² = u² + 2as. Option D (s = vt - at²) is incorrect — the correct form using final velocity is s = vt - ½at². All three correct equations can be derived from the definition of uniform acceleration.

In projectile motion (ignoring air resistance): horizontal velocity = u cos θ = constant (no horizontal force). Vertical velocity: changes due to gravity (increases downward, decreases going up). At highest point: vertical component = 0, but horizontal = u cos θ (unchanged).

72 km/h = 72 × (1000/3600) = 20 m/s. a = (v - u)/t = (20 - 0)/10 = 2 m/s². Always convert km/h to m/s first (divide by 3.6 or multiply by 5/18) before applying kinematic equations.

Inertia is the tendency to resist changes in state of motion. It is measured by mass — greater mass means greater inertia. Weight is force (mg) not inertia. A body at rest or moving uniformly has inertia proportional to its mass. This is the basis of Newton's First Law.

Gun recoil: bullet pushed forward (action) → gun pushed backward (reaction). By Newton's 3rd Law, forces are equal and opposite. By conservation of momentum: initial momentum = 0; final momentum of bullet + gun = 0 → gun recoils with momentum equal and opposite to bullet.

R = u²sin2θ/g; H = u²sin²θ/2g. For R = H: sin2θ = sin²θ/2 → 2sinθcosθ = sin²θ/2 → 4cosθ = sinθ → tanθ = 4 → θ ≈ 76°. This specific angle (tanθ = 4) is a standard CDS question.

Instantaneous velocity: velocity at a specific instant = limit of (Δs/Δt) as Δt → 0 = ds/dt (derivative of displacement). Average velocity = total displacement/total time. Speed = distance/time (scalar). Instantaneous speed = magnitude of instantaneous velocity.

A body is in equilibrium (translational) when the vector sum of all forces acting on it is zero (ΣF = 0). This means the body either remains at rest or moves with constant velocity. For rotational equilibrium, the net torque must also be zero.

Momentum p = mv = 2 × 5 = 10 kg·m/s. Momentum is a vector quantity in the direction of velocity. SI unit = kg·m/s = N·s. Newton's 2nd Law: F = dp/dt (rate of change of momentum). The impulse-momentum theorem: FΔt = Δp.

Time of flight for horizontal projection: h = ½gt² → t = √(2h/g). This depends only on height h and g — NOT on horizontal speed. Horizontal speed only determines horizontal range = v × t. All horizontally projected stones from the same height take the same time to reach the ground.

Area under a-t graph = ∫a dt = change in velocity (Δv). Area under v-t graph = displacement. Slope of v-t graph = acceleration. Slope of s-t graph = velocity. These fundamental graph relationships appear in every CDS Physics paper.

At the topmost point of vertical projection: velocity = 0 (momentarily at rest). Acceleration = g = 9.8 m/s² downward (gravity never stops acting). This is the most common misconception — students think acceleration is zero at the top. Only velocity is zero, not acceleration.

Impulse = Force × time = 20 × 4 = 80 N·s. Impulse = change in momentum: Δp = 80 kg·m/s. Alternatively, impulse = m × Δv = 5 × 16 = 80 N·s (Δv = Ft/m = 80/5 = 16 m/s). Impulse is a vector in the direction of force.

Galileo's law of free fall: all bodies fall with the same acceleration g regardless of mass (ignoring air resistance). Both reach the ground simultaneously. This was demonstrated at the Leaning Tower of Pisa. v = √(2gh) and t = √(2h/g) — both independent of mass.

Clock hands move at constant angular speed in circles — uniform circular motion. They have constant speed but changing direction (centripetal acceleration toward centre). The hour hand completes one revolution in 12 hours; minute hand in 60 minutes.