seegongsik

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 / 14

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

#ceiling fan#carousel#hard disk

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

Was this helpful? Support seegongsik