Running Jump and Air Resistance

Motion types

Translation and rotation

Equation 1D motion

x = x0 + v0t + 1/2at^2

Equation 2D motion

x = vx(0)t, y = vy(0)t - 1/2gt^2

Newton law

F = ma

Vertical jump

F = 2W, c = 0.6m → v = 3.4m/s, h = 0.6m

Running jump

Kinetic energy → gravitational potential energy

Jump height

H = v^2 / (2g)

Max range

R = v^2 sin(2θ) / g

Max height

ymax = v^2 sin^2(θ) / (2g)

Optimal angle

45° gives maximum range

Projectile motion

a = -g

Free fall

a = g, constant acceleration

Launch angle

tanθ = vy/vx

Running speed

10 m/s → 6.6 m jump height

Standing jump

Resultant force 1.16W, angle 65.7°

Range standing

R = 1.4m (record 3.71m)

Mechanical work

Weight × height

Energy efficiency

≈20%

Jump energy

70kg, 0.6m → 411J per jump

Total work

24.7×10⁴J in 10 min

Energy consumed

≈294 kcal

Air resistance

R opposes motion, ∝ v or v²

Drag form

Fa = CAv²

Drag constant

C = 0.88 kg/m³

Terminal velocity

vt = sqrt(mg / CA)

Skydiver

70kg, A=0.2m² → vt=62.4m/s

Parachute open

A=10m² → vt=8.8m/s

Bug fall

1cm bug → vt=8.6m/s

Hailstone

1cm → 8.3m/s; 4cm → 16.6m/s

No air resist

1cm hailstone → 140m/s

Weight relation

W ∝ L³

Area relation

A ∝ L²

vt relation

vt ∝ L

Resistive force

F = mg - bv

Linear motion

Uniform and accelerated

Projectile assumption

Neglect air friction

Terminal condition

Fa = mg

Acceleration zero

a = 0 at terminal velocity

Skydiver motion

Acceleration decreases until vt

Parachute effect

Upward acceleration after open

Translational motion

Rigid body moves without rotation

Rotational motion

Movement around axis

Centripetal force

Fc = mv²/r

Moment inertia

I = Σmr²

Rotational law

τ = Iα

Pendulum period

T = 2π√(L/g)

Physical pendulum

Uses moment of inertia

Walking model

Inverted pendulum

Running posture

Combination of translation + rotation

Muscle power

Muscles convert chemical → mechanical energy

Drag direction

Opposite motion direction

Body efficiency

Low; most energy lost as heat