Dynamics
Dynamics extends statics to bodies in motion, linking force to acceleration through Newton's second law. Problems split into kinematics (describing motion without asking why) and kinetics (relating forces, energy, and momentum to that motion). A handful of constant-acceleration equations and three conservation principles cover most FE questions.
Kinematics of constant acceleration
When acceleration a is constant, three equations describe the motion: v = v₀ + a·t, x = x₀ + v₀·t + ½·a·t², and v² = v₀² + 2·a·(x − x₀). Starting from rest at 3 m/s² gives v = 12 m/s after 4 s. The third equation is the go-to when time is neither given nor wanted: a car from rest at 2 m/s² over 25 m reaches v = √(2·2·25) = 10 m/s. Keep signs consistent — choose a positive direction and apply it to every vector quantity.
Projectile motion
A projectile's horizontal and vertical motions are independent and share only time. Horizontally, acceleration is zero so x = v₀·cos θ·t; vertically, gravity acts so y = v₀·sin θ·t − ½·g·t². A ball thrown straight up at 20 m/s reaches its peak when v = 0, at t = v₀/g = 20/9.81 ≈ 2.04 s, rising h = v₀²/(2g) = 20²/(2·9.81) ≈ 20.4 m. For a launch at angle θ over level ground, the range is R = v₀²·sin(2θ)/g, which is maximized at 45°: 20 m/s at 45° gives R = 400/9.81 ≈ 40.8 m.
Kinetics: Newton's second law
The net force on a body equals mass times acceleration, ΣF = m·a; a 10 kg mass accelerated at 5 m/s² needs 50 N. For rotation the analog is ΣM = I·α, where I is the mass moment of inertia and α the angular acceleration. Solving a kinetics problem means drawing the free-body diagram, summing forces (or moments) along the direction of motion, and setting the sum equal to m·a (or I·α). On an incline, resolve weight into components along and normal to the surface before summing.
Work and energy
The work–energy principle states that net work done on a body equals its change in kinetic energy: W_net = ΔKE, with KE = ½·m·v². A 10 kg mass at 4 m/s has KE = ½·10·16 = 80 J. Gravitational potential energy is PE = m·g·h, and a spring stores ½·k·x². When only conservative forces act, mechanical energy is conserved (KE + PE = constant), which turns many height-and-speed problems into one algebra step, bypassing the need to track time or acceleration.
Impulse and momentum
Linear momentum is p = m·v; a 5 kg object at 10 m/s carries 50 kg·m/s. The impulse–momentum theorem, F·Δt = Δp, links a force acting over time to the change in momentum. When no external impulse acts on a system, total momentum is conserved — the foundation of collision analysis: m₁·v₁ + m₂·v₂ = m₁·v₁' + m₂·v₂'. In a perfectly inelastic collision the bodies stick and move together; kinetic energy is not conserved, but momentum still is. Distinguishing which quantity is conserved is the key decision in impact problems.
Rotation and normal (centripetal) acceleration
A body moving in a circle of radius r at speed v has centripetal acceleration a_n = v²/r directed toward the center, requiring a net inward force F = m·v²/r. This governs banked curves, orbiting masses, and rotating machinery. For rigid-body rotation, v = ω·r and a_t = α·r relate linear to angular quantities, and rotational kinetic energy is ½·I·ω². Recognizing that circular motion at constant speed still involves acceleration — because direction changes — is a frequent conceptual test point.
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