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dynamics · how motion works

Motion, seen through calculus

From position to velocity to acceleration: one idea, the rate of change, runs through the motion of particles and rigid bodies.

14 lessons

01Velocity Is the Slope of Position, Acceleration the Slope of Velocity

From position x(t), differentiate to v = dx/dt and a = dv/dt, integrate to go back

#speedometer#GPS tracking#dashboard

02With Constant Acceleration, v and x Follow Simple Formulas

Constant a → v=v0+at, x=x0+v0t+½at², v²=v0²+2a·Δx, free fall (a=g)

#free fall#braking distance#drag racing

03A Projectile Is Horizontal Constant Velocity Plus Vertical Free Fall

Horizontal x=vx·t and vertical y=vy·t-½gt² superposed; range and peak

#basketball shot#cannon#water fountain

04Acceleration in Curved Motion Splits into Tangential and Normal

Velocity is tangent; a_t=dv/dt changes speed, a_n=v²/ρ changes direction (centripetal)

#roller coaster#highway curve#race track

05Net Force Makes Acceleration, ΣF = ma

Net force ΣF=ma, free body diagram, mass as inertia a=F/m, incline a=g sinθ

#elevator#playground slide#tow truck

06Work Equals the Change in Kinetic Energy

Work W=F·d (variable force = F-x area), W=ΔKE, KE=½mv², power P=F·v

#archery bow#weightlifting#car engine

07With Only Conservative Forces, Kinetic Plus Potential Energy Stays Constant

PE mgh, KE+PE constant, pendulum exchange, v=√(2gh), non-conservative loss

#roller coaster#playground swing#pendulum clock

08Impulse Equals the Change in Momentum

Impulse J=F·Δt=Δp, momentum conservation, F=J/Δt (airbag), collision

#airbag#boxing punch#rocket thrust

09A Collision Keeps Momentum, and the Restitution Coefficient Sets the Bounce

Momentum m₁u₁+m₂u₂=m₁v₁+m₂v₂, restitution e=separation/approach, elastic vs inelastic, KE kept (1+e²)/2

#billiard balls#car crash#bouncing ball

10Rotational Motion Is Described by Angular Position, Velocity, and Acceleration

Angular position θ, ω=dθ/dt, α=dω/dt, constant α (ω=ω₀+αt, θ=ω₀t+½αt²), link v=rω, a_t=rα, a_n=rω²

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11General Plane Motion Is Translation Plus Rotation

Translation+rotation split, rolling v=rω (d=rθ), point velocity v_P=v_O+ω×r (bottom 0, top 2v), instant center IC, cycloid

#rolling wheel#bicycle#bowling ball

12Relative Motion Is Solved by Shifting the Reference Frame

Relative velocity v_B=v_A+v_{B/A}, two body points ω×r, relative acceleration α×r−ω²r, rotating frame v_rel, Coriolis 2ω×v_rel

#hurricane swirl#crosswind landing#boat on river

13Torque Makes Angular Acceleration, ΣM = Iα

Torque M=F·d, moment of inertia I=Σmr², rotational Newton ΣM=Iα, shapes βMR² (hoop 1, disk ½, sphere ⅖), parallel axis I=I_cm+Md²

#wrench#revolving door#flywheel

14Rotation Has Kinetic Energy Too, ½Iω²

Rotational KE ½Iω², torque work W=Mθ (P=Mω), rolling KE=½mv²(1+β), energy conservation mgh=½mv²(1+β), finish speed by shape

#flywheel#ball down ramp#yo-yo

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