Motion and Force

Kinematics in one dimension

For uniform acceleration a, the four equations to memorise from FSc Punjab Textbook Chapter 3 (and HRW Chapter 2) are: v = u + at, s = ut + ½at², v² = u² + 2as, and s = ½(u+v)t. A car decelerating from u = 20 m/s to rest over s = 50 m needs a = (0 − 400)/(2·50) = −4 m/s². Free fall takes g = 9.8 m/s² downward; a stone dropped from a 45 m cliff hits the ground in t = √(2s/g) = √(90/9.8) ≈ 3.03 s with v = gt ≈ 29.7 m/s.

Newton's three laws

First law: a body remains at rest or in uniform motion unless acted on by a net external force (defines inertial frames). Second law: F_net = ma, or more generally F = dp/dt. Third law: every action has an equal and opposite reaction, applied to different bodies. The MDCAT classic: a 70 kg person in a lift accelerating upward at 2 m/s² feels apparent weight N = m(g+a) = 70(11.8) = 826 N; downward acceleration reduces it to m(g−a).

Projectile motion

Horizontal and vertical motions are independent. For launch speed u at angle θ: range R = u² sin 2θ / g, time of flight T = 2u sin θ / g, maximum height H = u² sin² θ / 2g. R is maximum at θ = 45°. Two complementary angles (e.g. 30° and 60°) give the same range. A ball thrown at 20 m/s at 30°: H = 400·0.25/19.6 ≈ 5.1 m, R = 400·sin 60°/9.8 ≈ 35.3 m.

Momentum, impulse, and collisions

Linear momentum p = mv is conserved when the net external force is zero. Impulse J = FΔt = Δp. In a perfectly elastic collision both momentum and kinetic energy are conserved; in inelastic collisions only momentum is conserved. For a 1-D elastic collision between m₁ with velocity u₁ and m₂ at rest, v₁ = ((m₁−m₂)/(m₁+m₂))u₁ and v₂ = (2m₁/(m₁+m₂))u₁. Equal masses simply exchange velocities — a billiard-ball result loved by examiners.

Friction and inclined planes

Static friction f_s ≤ μ_sN and kinetic friction f_k = μ_kN, with μ_s > μ_k. On an incline of angle θ, the component of gravity along the slope is mg sin θ and normal to it is mg cos θ. A block slides if tan θ > μ_s; on a frictionless incline the acceleration down the slope is g sin θ regardless of mass. Always draw the free-body diagram before plugging numbers — a habit endorsed by Serway.